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VALUE OF LIQUIDITY IN FINANCIAL MARKETS By VINAY DATAR A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1994 ACKNOWLEDGEMENTS My sincere gratitude for the invaluable guidance, encouragement and support of my dissertation committee cannot be sufficiently articulated. This intense effort would not have been possible without the understanding and support from my family and I shall, forever, remain grateful. TABLE OF CONTENTS ACKNOWLEDGMENTS . .* .* .* .* .* .* 9 9 9 9 9 9 9 9 3 ii ABSTRACT CHAPTERS A TUTORIAL ON LIQUIDITY Introduction * 3 1 3 9 9 9 9 9 9 9 9 9 9 9 9 1 . P 3 9 9 9 9 9 9 9 9 1 What s Liquidity? Overview of Literature Why do people demand liquidity? . Why is it costly to supply liquidity? * 9 9 9 9 3 . . 3 9 9 9 9 9 9 9 9 9 5 Incomplete markets . Imperfect markets . Market microstructure Different measures of liquidity Value of Liquidity: A Basic Model The economy * 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 3 9 5/ * 9 9 9 S 9 9 9 9 9 9 9 3 9 9 9 9 #7 S 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 S 9 9 * 9 9 a a 3 9 9 9 3 3 9 9 9 9 12 9 9 9 9 9 9. 9 3 9 9 9 9 9 9 9 9 9 9 9 9 9 9 1. Price of the Price of the Discussion 'liquid' 'illiquid asset asset a a 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 14. 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 91 &14 . 1 Summary CROSSSECTION OF STOCK RETURNS REVISITED LIQUIDITY PREMIA AND ROLE OF SIZE Introduction . .. .. 23 S9 9 9 2 Data and Methodology 9 9 S 3 9 9 9 9 9 9 9 9 9 9 9 9 9 9 3 9 4 9 9 .2 Description of data Estimation of beta Proxies for liquidity * 9 9 9 9 9 * 9 9 9 9 * * 9 9 0 3 * 9 9 9 .9 * 9 9 2 * 9 9 2 * 9 9 2 1 _ _I 1 I Summary . S 40 IMPACT OF LIQUIDITY ON PREMIA/DISCOUNTS IN CLOSEDEND FUNDS Introduction a a a a a 50 a a a 50 Liquidity and Premia/Discounts . . How is liquidity related to premia/discounts Testable hypotheses Proxies for liquidity Data and Methodology Data . . * C C C U C 53 * C S 53 S S S S U C b C C S C S C S C S S C C C C 5 4I . . 56 * C S . a a C S a a S S C C C C C C C 5 .5 6 * C C C C U C S C C C C S 4 5 C C S C C C S C .5 6 M ethodology ..............................58 Discussion of Results . . . . . . . .59 Sensitivity Analysis . . . . . . . 62 Some Informal Evidence . . .. . . . .65 67 APPENDIX A APPENDIX B PROPERTIES OF PORTFOLIOS TESTABLE HYPOTHESES: AN ALGEBRAIC ILLUSTRATION . . . 85 REFERENCES .88 BIOGRAPHICAL SKETCH .93 Summary Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy VALUE OF LIQUIDITY IN FINANCIAL MARKETS By VINAY DATAR APRIL Chairman: Dr. Robert C. Radcliffe 1994 Major Department: Finance, Insurance and Real Estate In ideal markets, asset prices depend purely on fundamentals and asset liquidity is not a concern because buyers (willing to pay the fundamental value) can be readily found. However, when markets are less than ideal, liquidity considerations may affect equilibrium asset prices, reflecting the anticipated frictions in trading. It is difficult to formalize liquidity because it may be influenced by a variety of trading frictions related to various forms of market imperfections or incompleteness. Chapter 'tutorial' on liquidity It briefly reviews different types of trading frictions that may affect liquidity of traded assets. Several, albeit noisy, measures of liquidity are outlined. A simple model of liquidity is presented to demonstrate the basic economics of liquidity. If liquidity has value then investors may support for such liquidity premia in asset returns? Chapter 2 examines this question and presents evidence of liquidity premia in the crosssection of stock returns. It is suggested that the well known relationship between firmsize and returns may, in fact, be due to liquidity considerations. Further, the relationship between returns and liquidity is found to be robust to any potential seasonality in stock returns (i.e., the 'January effect'). Chapter 3 examines the impact of liquidity on the pricing of Closed End funds. Funds can potentially create value by buying illiquid assets and selling more liquid claims. The claims issued by such funds would be more valuable than the underlying assets and would, therefore, trade at premium relative to the underlying assets. The empirical evidence is consistent with this notion. Specifically, in a cross sectional study, premia increase (or discounts decrease) as liquidity of a fund increases (as measured by proxies). Overall, the results suggest that liquidity considerations may play an important role in the determination asset prices in financial markets, and may plausibly explain some of the wellknown anomalies in financial markets. CHAPTER A TUTORIAL ON LIQUIDITY Introduction The notion of asset liquidity lacks a precise formal definition despite its intuitive appeal. Grossman and Miller (1988) express this idea quite succinctly: Keynes once observed that while most of us could surely agree that Queen Victoria was a happier woman but a less successful monarch than Queen Elizabeth I, v would be hard put to restate that notion in precise mathematical terms. with equal force to ti 617). Keynes's observation could apply he notion of market liquidity. (page Lippman and McCall (1986) observe that academic economists do not have a definition of liquidity as a measurable concept, although there is a general agreement that liquidity is the 'marketability' of an asset; or the property of an asset that facilitates immediate exchange for cash, without affecting the market price. They define liquidity in terms of the time required to sell an asset to the best bidder (on average). Grossman and Miller (1988) suggest that liquidity is related to the 'price concession' that may be demanded by a potential buyer to participate in an immediate trade. Although a precise metric of liquidity remains elusive in theory, in practice 2 Stock Exchange) fact book (1990), the dollar volume of trade on U.S exchanges exceeds $ 2 trillion per year, where the total market value of (potentially tradable) listed assets is about $ 3 trillion. This amounts to an yearly turnover rate of about 70%. This suggests that investors do care about the ability to trade, and are willing to pay the attendant transaction costs. Further, the chaotic frenzy on the trading floors suggests a strong sense of urgency to trade. Why might so many traders fall over each other (often times literally), to frantically engage in trade? How might trading, or frictions in trading, affect long run prices? These questions form the central theme of this study; what is the meaning and value of liquidity? Liquidity considerations play no role in traditional asset pricing models; any amount of trade can occur in equilibrium, without affecting prices. This result is natural when markets are perfect. Here, the market value of liquidity is zero, because market participants can supply liquidity at zero cost, if and when liquidity is demanded. Recent literature on liquidity attempts to provide some insights into the impact of trading (and frictions in trading) on asset values. The general suggestion is that less liquid assets have higher expected returns (or lower prices) than more liquid assets.1 These higher returns, or liquidity premia, are a compensation demanded by market participants to offset anticipated costs of trading. Liquidity considerations provide valuable insights into asset pricing to the extent that they explain, albeit in part, some well known empirical anomalies: 3 1) Claims to identical cash flows may have different prices [ See, e.g., Amihud and Mendelson (1991), Boudoukh & Whitelaw (1991)]. 2) Small firms have higher average returns than large firms [e.g., Stoll and Whaley (1983)]. 3) Closedend funds trade at discounts or premia relative to the market value of underlying assets [See Chapter 3]. This chapter proceeds as follows: The first section describes various motives for trade, sources of potential frictions and prevalent measures of liquidity. second section presents a simple model that captures the basic economics of liquidity premia. The third section concludes. What Is Liquidity?: Overview of Literature It is helpful to think about liquidity and its value in terms of the supply and demand for liquidity. In particular, let us examine why people might want to trade (demand liquidity); and why it might be costly to supply liquidity. An examination of these aspects might suggest a clue as to the meaning and value of liquidity. Why Do People Demand Liquidity? Fundamentally, individuals would want to trade (i.e. demand liquidity) when private valuation for an extra unit of an asset is different from market valuation. In such situations there are gains to be made from trading, and this suggests a motive for 4 Literature has postulated several reasons for such a differential valuation, where private value may be different from market value. Unforeseen shocks to preferences, endowments, information (about asset quality) and lifecycle trading may lead to personalized values that are different from market value. For example, Amihud and Mendelson (1986), Diamond and Dybvig (1983) and Flannery (1991) consider a situation where individuals are hit by a preference shock; here some agents early in the sense that such agents care only about current consumption, and they have no value for future consumption. Clearly, these agents would be willing to sell (to the highest bidder) at any nonnegative price. In similar spirit, Campbell, Grossman and Wang (1992) introduce change in the degree of risk aversion as a motive for trade; here, personal values are affected because of the change in risk aversion and agents can gain by trading. Admati and Pfleiderer (1988), Glosten and Milgrom (1985) and Kyle (1985) present models with private information as a motive for trade; smart traders have private information about the true value of the asset, and such traders can benefit by trading, as long as the market value is different from the true value. DeLong, Shleifer, Summers and Waldman examine the impact of 'noise'; irrational traders perceive the market value to be too low or too high, and such traders arrive in the market as buyers or sellers. Constantinides (1986) and Grossman and Miller (1988) introduce rebalancing as a motive for trade; individuals trade to allocate optimal amounts of their wealth between risky and riskfree assets. In such 5 individual would hold; trading occurs until, in equilibrium, the private values of all agents are equal to the market value. In summary, the liquidity literature considers four main motives for trade: 1) private information about quality of assets (informed traders), 2) private beliefs about quality of assets (noise traders), 3) need for rebalancing to maintain optimal level of risky assets in the portfolio (liquidity traders) and 4) consumption needs (liquidity traders). Why is it Costly to Supply Liquidity? When markets are perfect and complete, trading can occur at zero cost, and as a result liquidity can be supplied at zero cost. should be zero. Therefore, the market value of liquidity The literature examines the issue of the cost of liquidity in two separate but related ways: 1) incomplete markets and 2) imperfect markets. In both cases, there are costs associated with trading, and investors demand compensation to offset the anticipated costs of trading. An increase in such anticipated costs, results in a lower exante price (or higher expected returns) for a given asset (or a claim to a given set of cash flows). The extra return due to liquidity considerations is called liquidity premium. Incomplete markets. Let's consider the incomplete market framework. When markets are incomplete, statecontingent claims are not available for some states of incomplete. Such an approach allows us to study the 'insurance' aspect of liquidity and the value of such an insurance. Consider the following; investors may be driven to sell (demand liquidity) because of an endowment shock, or because of an exposure to some other risk that is not insurable, and all investors are exposed to such a nondiversifiable risk. In such a situation, the investors may need to worry about an additional risk (in addition to fundamental risk) because the expected resale price may be sensitive to trading (i.e. doesn't depend purely on the asset fundamentals). For example, Campbell, Grossman and Wang (1992) introduce a personal shock to risk preferences as a nondiversifiable risk. Such shocks increase the volatility of prices. Risk averse rational traders would demand extra premium in such a market because they are not immune to the same potential shock in the future. This combination of nondiversifiable excess volatility and riskaversion gives rise to liquidity premia.2 Diamond and Dybvig (1983) show that illiquid assets have higher returns than liquid bank deposits because, in effect, the liquid deposits provide guaranteed consumption.3 In their model, productive investments are not reversible, and such investments have only one known terminal payoff with no intermediate dividends. Further, the claims on these investments are not tradable at intermediate dates (real results model e ra s i m i l ar to noise trader  ..  .  ..  ,  model presented V 7 investments are illiquid). All individuals in the economy are identical (on date 0), but a few of them (randomly) may 'die' early (before the terminal payoff is received); 'diers' need to consume early, therefore they would demand liquidity. On date 0, all individual know that they have some probability of being a demander of liquidity. Investors may pass up the illiquid investment, despite the high returns that it provides. A bank creates liquidity by issuing demand deposits and using some of the proceeds to invest in the illiquid productive asset. Investors are willing to accept a lower return on the liquid deposits (as compared to higher return on the illiquid productive asset).4 In summary, the mere possibility of any uninsured shock to beliefs, risk preference, timepreference, income, endowment, consumption liabilities or human capital is can potentially, give rise to liquidity premia.5 Imperfect markets. The liquidity literature considers a variety of market imperfections that may generate liquidity premia. Constantinides (1986) examines the impact of transaction costs on expected returns. In his framework, investors tradeoff the benefits of optimal rebalancing against the cost (due to brokerage costs) of trading; however he finds that the transaction costs have only a 'second order' effect on the risk premia. Intuitively, his results suggest that the need to rebalance may not be an important reason to demand liquidity; as a result, investors can tolerate any loss 4 Jacklin (1987) points out that the liquidity premia in such a model, with a risk neutral framework, crucially depend on market incompleteness (introduced here bv the 8 in utility due to suboptimally balanced portfolios and forego liquidity rather than pay transaction costs. Amihud and Mendelson (1986) suggest that investors do consider the present value of anticipated transaction costs at the time of initial purchase, this results in lower prices (or higher expected returns) for less liquid assets. Market microstructure. The emerging literature on market microstructure considers the details of the market mechanism that facilitates trade, and examines the attendant frictions in the trading process. In such models, the traders arrive at different (asynchronous) times, and a market maker stands ready to be a counterpart for trades. Such a view (of the market) enriches the traditional view of the market as a giant trading hall, where symmetrically informed traders submit their demand schedules and trade occurs, if at all, after the equilibrium price is determined. market maker buys at the bid price and sells at the ask price, from any willing trader. The difference between ask and bid provides fair compensation to the market maker, to offset the costs of making a market. The costs of market making may be broken down into three parts as 1) adverse selection costs [Glosten and Milgrom (1985)], 2) inventory management costs [Ho and Stoll (1981), Grossman and Miller (1988)], and 3) order processing costs [Roll(198, market maker, 4)]. The adverse selection costs refer to the losses incurred by the while trading with traders who have private information. In the process of providing immediacy, the market maker may be called upon to allocate a These costs are called the inventory management costs. Order processing costs are simply the administrative costs of maintaining the trading establishment. The bidask spread, that potentially offsets the costs of making a market, represents a friction in the trading process and is therefore viewed as related to liquidity. The quoted bidask spread is valid for a defined size of trade (usually around 500 shares) and larger orders may temporarily change market prices. of trade that may temporarily move prices by one dollar is called 'depth' The size and the speed with the price 'recovers' to some long run 'natural' level is called the 'resiliency'.6 The bidask spread, depth and resiliency collectively characterize market liquidity. Different Measures of Liquidity It is difficult to define a single metric for liquidity because liquidity is influenced by several different sources of trading frictions. Several measures of liquidity have been discussed in literature, some of which are reviewed below. The bidask spread has been suggested as a measure of liquidity by several authors because it is a mechanism that compensates the marketmaker who is the provider of last resort for liquidity [e.g., Demsetz (1968), Glosten and Milgrom (1985), Stoll (1989) and Amihud and Mendelson (1986)]. The market maker may simultaneously trade at the bid (for a buy transaction) and at the ask (for a sell transaction), while profiting from the spread. In a competitive market for marketmaking services, the expected profits to the marketmaker would be a fair measure of the cost of providing liquidity services. There are some problems with this argument. Lee, Mucklow and Ready (1993) argue that bidask spread, by itself, may be a misleading measure of liquidity because the quoted bidask spread is valid only for a limited trade size. They suggest that depth and bidask spread, together, provide a better measure of liquidity. In similar spirit, Grossman and Miller (1988) argue that bidask spread fails to measure the cost of immediacy (or the cost of delaying a trade); and further, the bidask spread may not be a fair measure of market maker's compensation, to the extent that the market maker may not simultaneously trade at both the bid price as well the ask price. Several authors have noted that the quoted bidask spread may not be the effective spread, because many trades may occur inside the bidask spread, and spread may be an extremely noisy (to the point of being meaningless) measure of liquidity. Liquidity ratio, defined as average dollar volume of trade, per unit price change over some interval, is sometimes used as a measure of liquidity [e.g., Copper, Groth and Avera (1985)]. Grossman and Miller (1988) point out that this measure fails to capture the impact of a larger than average trade, and further it does not account for fundamental volatility. For example, release of in this case, because the change in price is not caused by liquidity considerations, but purely by fundamental revaluation. The more efficient the market, the smaller would be the measure of liquidity ratio, thereby it would understate the true liquidity. Volatility ratio, defined as a ratio of long term return volatility to short term return volatility, is suggested by Barnea (1974) as a proxy for liquidity. This ratio attempts to capture the price volatility caused by the order imbalance, albeit imperfectly. A change in fundamental volatility can introduce noise to this measure, because it cannot be distinguished from liquidity related volatility. This ratio may be a reasonable measure of liquidity to the extent that fundamental volatility can be assumed to be stationary. Autocorrelation of returns may serve as a proxy for liquidity. Grossman and Miller (1988) and Goldman and Beja (1979) show that the serial correlation of returns reflects the degree of participation of the market maker in the trading process. Highly traded securities should have a lower (closer to zero) serial correlation of returns. Thinly traded securities may have a larger (away from zero) negative serial correlation of returns. The price smoothing function of the market maker may create some positive serial correlation, and the inventory management process may introduce a negative serial correlation. These opposing effects add noise to the measure but, a larger negative serial 12 Volume of trade is suggested as an indicator of liquidity some authors [e.g., Benston and Hagerman (1974); Stoll (1978); Barclay and Smith (1988)]. intuition is that high trading volume may mitigate the problem of inventory management for the market maker, and thereby reduce trading costs; this can occur if buy and sell orders arrive rapidly with equal probability, and the market maker can simply cross orders. Further, higher volume allows the fixed cost component of administrative costs to be spread over a larger number of shares, and the per share cost of trading is reduced. Higher volume of trade may indicate higher liquidity. The next section demonstrates the relationship between trading frictions and the value of liquidity, using a simple model. Value of Liquidity: A Basic Model In this section, we develop a simple model that provides a framework to study the basic economics of liquidity. An important contribution of this model is the demonstration that liquidity premia can arise in a fairly simple framework and that a richer framework, although enriching, is not essential for liquidity premia to exist. The simplicity of the model helps identify the bare essentials of liquidity and its relationship with long run value. We suggest that liquidity premia may arise even in a simple economy with dividend certainty, risk neutrality and exante homogeneity above assumptions (as modelled in related literature) creates a broader potential role for trading and may, therefore, further increase the equilibrium value of liquidity. The Economy Consider an exchange economy with only two assets: a 'liquid' asset and an 'illiquid' asset; both the assets represent traded claims to a perpetual stream of guaranteed dividends of $D per period. The amount and timing of dividend payout is known (to everyone) with complete certainty and in this sense both the assets are identical. 'liquid' asset is different from the 'liquid' 'illiquid' asset in the following sense: claim provides a guarantee of a resale price, P1, at any time in the future. Further, the supply of this asset is infinite and it cannot be sold short.7 For example, consider a redeemable U.S. Government consol that has a guaranteed redemption value but this consol cannot be sold short. In contrast, the resale price of 'illiquid' asset is not guaranteed.8 SThis restriction is necessary to impose illiquidity on the illiquid asset, otherwise everyone will prefer to short sell a liquid asset instead of selling an illiquid asset. In other words, allowing short sale of the liquid asset implicitly provides a costless way of disposing off the illiquid asset. Consider, for example, that U.S. treasury bills are liquid, but can not be sold short. The investors are identical exante and everybody is risk neutral. The riskfree discount rate is r%. A subset of investors may experience a shock to their time preference, at some future time. The exante probability, ir, of this event is common knowledge but the outcome is known only privately expost. This risk is not insurable because it cannot be observed publicly. Price of the 'Liauid' Asset The 'liquid' asset guarantees dividends as well as resale price. The amounts as well as the timing of cashflows from this asset are known with certainty. The risk neutral value of this consol is simply the present value of expected future cashflows discounted at the riskfree rate. D D L (l+r) (l+r)2 Where, L = the guaranteed redemption value of the liquid asset S= the dollar amount of the riskfree dividend paid periodically = the riskfree discount rate Price of the Illiauid Asset rWt1 rI I 1 t   P f a Where, = the market clearing price of the illiquid asset at any time, t, in the future. The initial price of the illiquid asset; and it is equal to the private valuation of individuals who, by chance, are not affected by a timepreference shock, at any given time t in the future. = some unspecified function of the illiquidity of the asset, at time t. In some sense the variable, , reflects the price concession demanded by the buyers. Tilde () denotes random variables and all the random variables above are restricted to be nonnegative. Further, let's assume that the distributions of all the random variables are stationary through time. Equation (2) says that the resale price, price, Po, net of any trading friction, is equal to the buyer's reservation The buyer's reservation price, in turn, mlr /r i 1 ,I . . ^ ^ L J  ^ ^1 t .li 1  E( P M) E( M) The present value of these payoff possibilities is equal to Po,, 7r) (D+Po) (1+r) (1 i) (D+Po) where 7 [D+E(PM)] (1+r) 7 [D+PoE(f)] (1 +r) (1+r) D r Tr E(f') _ ir E(f) r Equation (3) says that the initial price of an illiquid asset is lower than the price of an identical liquid asset by an amount equal to the present value of expected M fl an .A A I, *1,1 ... ZJ1 L_. _. .: 1 .5 5 D+ D+ D + D + 1 1 1 1 a liquid asset, by virtue of a guaranteed resale price; such an option is not available on an illiquid asset and the relative prices properly account for this aspect. In our story the buyer's reservation price, Po, is stationary through time (because the future prospects are unchanged); and it is not the same as the resale price, , (or the market clearing price) in the future. Under these conditions, there is further gain from trading and this is counter to the notion of an equilibrium. In order to sustain these divergent prices in equilibrium, we need some friction to restrict further trading.9 It is this friction that captures the essence of the cost of liquidity at the margin. For example, it can be seen from equation (3) that in a frictionless world, E(j)=0, the price of an illiquid asset is the same as the price of a liquid asset. This makes sense because, by assumption, liquidity is freely supplied. We discuss the interpretation of this friction, later on. We demonstrated that liquidity premia, higher returns10 or lower price, may arise even in a simple economy with exante homogeneity, risk neutrality and dividend certainty. All that is needed to justify liquidity premia is some friction in the trading process (and some motive to trade). In essence, market imperfection/incompleteness generates liquidity premia. 18 In our framework, investors' trading motive was generated by an uninsurable shock to the timepreference, and the trading friction was in the form of sellers receiving a price below the buyer's reservation price. Discussion: Let's consider extreme values of the trading friction, , to develop some feel for its impact on prices. At one extreme, if we eliminate the trading friction, the liquidity premia disappear. This can be seen from equation (3) by setting the value of E(f) equal to zero. In other words, without trading frictions, all traded assets would be fully liquid and their prices would be identical, for a given stream of dividends. At the other extreme, if the trading friction were to be very large then the market would not open and the resale price of the asset would be zero (the smallest possible price). From equation (3) we have: D (r+7r) This is analogous to a complete market failure, at the time of 'death', probability of 'death' and the liquidation value is zero. where 7r is the We would expect the trading friction to be in between these two extremes and this suggests a range for the value of liquidity. w I I I  unchanged. The market clearing price, may change from time to time depending on the trading friction, ', that is prevalent in the market at a given time. proceeds of sale received by the seller, are exactly equal to the buyer's reservation price net of the trading friction, in other words the seller bears all the costs associated with selling (and the exante estimate of this cost is reflected in the buyer's reservation price, Po, because at some time in the future the buyer may become a seller). We interpret the friction, , as the pershare cost of operating the market at any given time. If the cost of operating the market is fixed then the per share cost would be lower for an asset with a higher trading volume, because the fixed cost is spread out over a larger number of traded shares, at any given time. suggests that across different assets, the market price, This would be increasing the higher the volume of trade. Alternatively stated, the observed returns, would be decreasing the higher the trading volume. Further, this negative relation between returns and volume gets stronger if we look at returns over longer holding periods (say over n periods); because the random resale price, becomes less important and the dividends (n times D) play a larger role in the observed returns, (PR +nD)/P  'WI This result has an easy intuitive interpretation; investors can get higher +D)/ , 20 The essence of value of liquidity can be easily described in the context of the model above. Investors know that they may have a liquidity need in the future, and they consider the expected future resale price at the time of initial purchase. If the expected future resale price depends purely on fundamentals then initial purchase price also depends purely on fundamentals; and the market value of liquidity is zero. However, if the future resale price is expected to be less than the fundamental price then initial purchase price is reduced to that extent; and in equilibrium the deviation from the fundamental value reflects the impact of liquidity considerations. Liquidity considerations become important when there is a 'wedge' between the expected proceeds to the seller and the reservation price of the buyer. An intuitive interpretation of such a 'wedge' may be as follows: when firms are liquidated, the liquidation proceeds may be lower than the going concern value. The difference between the going concern value and the liquidation proceeds is an example of a 'wedge' In financial markets, such a wedge can be caused by a variety of frictions to trade; in our model such a wedge (t) was exogenous. Explicit considerations of the details of the market mechanism, asset characteristics and the nature of the economy may give rise to this wedge (f)." For example, brokerage costs, information asymmetry limitation on risk bearing capacity etc. can potentially create frictions (i.e. the 'wedge' or 'f') that result in liquidity being costly to supply and therefore valuable. 21 Summary The meaning and value of liquidity can be understood by looking at the demand for and supply of liquidity; namely why do people trade? and why it might be costly to facilitate trade. Literature suggests four main motives for why people trade (or demand liquidity): 1) private information about asset quality (information motive), 2) private beliefs about asset quality (noise motive), 3) need for rebalancing to maintain an optimal level of risk in the portfolio (rebalancing motive) and 4) life cycle needs or cash needs to meet consumption liabilities (consumption motive). On the supply side, it may be costly to supply liquidity (facilitate trade) because of three main reasons: processing costs. 1) adverse selection, 2) inventory costs and 3) order The model in the second section demonstrates, that the cost of supplying liquidity creates a 'wedge' between the reservation price of the buyer and the proceeds of sale received by the seller. This 'wedge' is the essence of the market value of liquidity. An explicit consideration of liquidity (the market value of liquidity), affects ante prices of traded assets; i.e. prices depend on fundamentals as well as on the cost of liquidity For a given set of fundamentals, assets that are more costly to trade (less liquid assets) are worth less; this reflects the market value of liquidity Alternatively 22 over the fundamental returns (premium over hypothetical returns in absence of liquidity considerations) is called the liquidity premium. In essence, trading frictions impair liquidity; and liquidity premia arise as a compensation for anticipated frictions in trading. Liquidity may reflect a composite influence of several types of trading frictions, and therefore it is difficult to devise a single metric for liquidity. The bidask spread, volatility ratio, market depth, auto correlation of returns and volume of trade provide a way of quantifying liquidity of a traded asset. Liquidity considerations may be the bridge between the perfect world of theory and the clumsy imperfect world in practice. Liquidity has important implications that potentially explain, albeit in part, some of the well known anomalies in financial markets. The next two essays empirically examine the impact of liquidity premia on traded assets. The first essay examines all the stocks traded on the NYSE and AMEX stock exchanges. The evidence suggests that the crosssectional variation of returns may be related to liquidity. The second essay suggests that the premia and discounts in closedend funds may be influenced by liquidity considerations. CHAPTER CROSSSECTION OF STOCK RETURNS REVISITED: LIQUIDITY PREM1A AND ROLE OF SIZE Introduction The well known relationship between size and returns achieved greater prominence in a recent article by Fama and French (FF) (1992). size plays a major role in describing the crosssection of returns, completely fails to explain any variation in returns. They suggest that whereas CAPM just The economic interpretation of the role of size is an open issue. FF (1992) do not make any claims as to the underlying economics, but provide a conjecture that size may capture some risk. In similar spirit, Berk (1992) and Berk and Takezawa (1993) suggest that market value, to the exclusion of physical size attributes (operating measures like sales, number of employees, etc.), captures risk and therefore should be related to returns. The notion that size captures some risk is appealing at first glance. as shown by Keim (1983) and further confirmed by this study relationship is not observed in nonJanuary months. However, the sizereturn The notion that size captures some unobservable risk is not very useful if size is not even related to returns most of the time (or in nonJanuary months). 24 Literature on liquidity' suggests that impediments to liquidity may result in higher equilibrium returns for illiquid assets (i.e. liquidity premia).? In this spirit, James and Edmister (1983) offer the insight that the observed relationship between size and returns may, in fact, reflect a more fundamental relationship between liquidity and returns; however, in their data they do not find any support for their conjecture. In this study, using a broader data set, we explore this conjecture and find support for the notion that size may, in fact, reflect liquidity. The main objective of this study is to investigate any influence of liquidity on stock returns. We find that the empirical results strongly support the notion of liquidity premia. Specifically, thinly traded (low trading volume) stocks provide higher average returns than highly traded (high trading volume) stocks. In order to clarify the role of size visavis volume, size and volume. we examine portfolios sorted on the basis of The relationship between volume and returns persists even after controlling for size. Further, upon controlling for volume, the influence of size on returns is insignificant. Moreover, in nonJanuary months, size has no relationship example, Amihud Mendelson (1986) Bhardwaj Brooks (1992), Boudoukh and Whitelaw (1993), Constantinides (1986), Demsetz (1968) Diamond and Verrecchia (1991), Flannery (1991), Grossman and Miller (1988), Jacklin (1987), Pagano (1989), Reinganum (1990), Shleifer and Vishny (1992), Stoll and Whaley (1983), Tinic (1972) examine various impediments to trade and suggest a role for liquidity in asset returns. This a partial list and certainly not exhaustive. 25 with returns but volume continues to explain the crosssection of returns. Another proxy for liquidity (the number of shares outstanding) shows essentially similar results. This suggests that, the relationship between size and returns may reflect a more fundamental relationship between liquidity and returns. We do not pretend to have explained the January effect. Size does have a very strong relationship with returns and it is limited to January. Our results suggest that the observed overall relationship between size and returns seems to be subsumed by liquidity proxies, and that the influence of liquidity is not limited to January. In summary our findings suggest three important conclusions: (1) liquidity premia do indeed exist in practice and they are pervasive (not limited to January), (2) the observed overall relationship between size and returns may be due to liquidity premia, and (3) liquidity premia may play a major role in determination of relative asset prices to the extent that size plays a major role per FF (1992). In the first section, we describe the data, our method of estimating beta and proxies for liquidity. In the second section, using the same methodology as used by FF, we examine the relationship between average returns and size as well as volume and another liquidity proxy. We find that liquidity seems to have a stronger relationship with returns than size does. This finding is further confirmed by the Davidson and MacKinnon's Jtest for nonnested hypotheses. The third section provides additional evidence in support of liquidity premia using test portfolios to 26 but the liquidity effect is not limited to January. In the fourth section, these findings are further corroborated by the analysis of residuals using a methodology similar to Chen (1983). The final section summarizes and concludes. Data and Methodology Description of Data The data set includes all firms on the NYSE exchange from the Center for Research in Security Prices (CRSP). Monthly data is collected from the CRSP tape, from July 1962 through December The time period was selected to be essentially similar to the time period selected by Fama and French (1992). Log of firm size (LSZ.,_) is defined as the natural logarithm of total market capitalization at the end of the prior month (or month tl) for a given firm. Log of volume of trade (LVOL1..1) is defined as the natural logarithm of the average monthly volume of shares traded in the prior 3 months (or month t3 to month t1).3 The log of outstanding shares (LSH.1 ) is defined as the natural logarithm of the number of shares outstanding at the end of the prior month (or month t1). Returns are expressed as a percentage change in the value of one dollar of investment over the time period of interest. We examine returns over three different horizons; monthly, sixmonthly and yearly. 27 Monthly returns are simply the returns in the current month (or month t). Yearly returns are calculated as nonoverlapping holding period returns. These are the returns for the current and the following eleven months; and reflect the percentage change in value of one dollar invested over the time period. After the yearly return is calculated for the current observation month (or month t), the next observation for the firm is after 12 months (or month t+12). Sixmonthly returns are calculated similar to the nonoverlapping yearly returns. Using monthly returns does not provide an explicit role to the actual dividend stream, whereas using longer term returns does allow an explicit role to the dividends.4 The cost of using long term returns is that we get fewer observations, and implicitly we assume stationarity of the attributes under investigation. Using overlapping returns is one way around the problem of fewer observations, but this may overstate the tvalues to the extent the observations are not entirely independent, we do not consider overlapping returns in this study. 4 It is important that the actual dividend stream be allowed to play an explicit role in calculation of returns because we want to examine the longrun effect of liquidity on a given stream of dividends. Prices may permanently deviate away from the 'fundamental value' to compensate for illiquidity and this is the aspect that we want to investigate. For example a consol of $ per period valued at $10 provides an implicit 10% return, and the same console provides a 20% return if valued at $ 5. In reality, fundamental values Estimation of Beta Fama and MacBeth (1973) and Fama and French (1992) use the portfolio approach to estimate betas in an attempt to reduce the noise in the estimate of betas. Fama and French (1992) find that any beta measurement errors do not affect the ordering of betas, and the portfolio approach is unnecessary for beta estimation. Further, beta does not describe the crosssection of returns in their data. We expect similar results because our sample and methodology are essentially similar. We do not follow the portfolio approach, but instead, simply calculate betas for individual stocks. Since monthly returns are used, nonsynchronous trading does not present a major problem, so adjustment suggested by Dimson (1979) is not necessary. values are calculated for each firm, in each month, using past 30 to 60 months of data (based on availability) and using the value weighted portfolio of NYSE stocks from the CRSP tape as a proxy for the market portfolio. Proxies for Liquidity The literature on liquidity recognizes that liquidity may have value as long as markets are incomplete or imperfect in some sense. Transaction costs, information asymmetry or heterogeneity (of endowments, preferences or investment horizons) may generate liquidity premia.6 However, there is no consensus about a good metric for liquidity. For example, Amihud and Mendelson (1986) use the bidask spread to measure the cost of immediate execution or the liquidity of an asset, while Stoll (1985) and Grossman and Miller (1988) argue against bidask spread as a measure of liquidity (see, Chapter I), in spite of its rough common sense appeal as a metric for liquidity. They contend that, in general, the bidask spread fails to capture the market makers' return for providing immediacy which is the cost to the investors for receiving immediacy. In the context of alternative proxies for liquidity Grossman and Miller (p. 630) argue that 'what we need is a measure of how well the market makers are providing customers with an effective substitute for the delays in a search for a more inclusive set of counterparties'. They suggest that the ease with which one can find a counterpart to a trade is an important attribute describing the liquidity of an asset. In the spirit of Grossman and Miller, we use the trading volume of a stock (i.e., the average number of shares of a stock traded over the previous year), as a measure of the liquidity of that stock. The idea being, the higher the trading volume of a stock, the easier it is to find a counterpart to trade with and hence, the greater the liquidity of that stock and vice versa. Higher trading volume, to the extent it indicates that there are fewer impediments to trading, may be a good proxy for liquidity. We use monthly volume of trade, averaged over the prior three months, as outstanding shares is a variable of interest because in Merton 's (1987) model, it explains crosssectional differences in stock returns. In fact, Merton (p.494) calls it the degree of 'investor recognition' or the relative size of investor base. In general, since the stocks having a larger investor base are also traded more frequently, the number of shares outstanding can also be closely linked to the marketability or liquidity of a stock. Returns, Size and Liquidity In this section we use the methodology used by FF (1992) and examine the influence of size and liquidity proxies on the crosssection of returns. The analysis uses the basic methodology developed by Fama and MacBeth (FM) (1973) to test the null hypothesis of no relationship between the explanatory variables and returns (hypothesized dependent variable). The FM regressions are carried out as follows: starting with January 1963, perform a crosssectional regression of returns on the explanatory variables. Such a regression is carried out every month up to, and including, December This provides 348 monthly estimates of slopes for each of the explanatory variables. Average slope is the arithmetic average across the time series of 348 monthly slope estimates. Associated tstatistic is simply the average slope divided by its time series standard error across the 348 monthly estimates of slope.7 The average slope in FM 31 tests represents the average effect of each explanatory variable on returns (the hypothesized dependent variable). Table I summarizes the results of monthly FM regressions. Size has an average slope of 0.14 and the associated tvalue is 2.58, this is about the same as reported by FF (1992). Liquidity proxies, volume and number of outstanding shares, seem to show a somewhat stronger statistical relationship with returns than size does. For example, volume has an average slope of 0.12, with a tvalue of 2.95. LSH (number of outstanding shares) show an average slope of 0.16, with a tvalue of 3.13. However, the explanatory power of size as well as liquidity is not significant when these variables are used simultaneously. This may be because of multi collinearity in the data and the attendant lack of precision in the estimates. As shown in Table II, size and liquidity variables are highly correlated and this may, potentially, create the usual difficulties of interpretation in a multivariate analysis. Later on, in the third section, we use test portfolios to study the independent effect of size and liquidity 7(...continued) stationary, slope esti mates are assumed to be serially independent. alternative aggregation method (viz. Fisher test) , does not rely on stationarity, but does assume timeseries independence. The Fisher test aggregates 'pvalues' for the timeseries of slope estimates, to develop a statistic that is distributed as Chi square. In our data, the results of the Fisher test are in agreement with the results from the FM methodology. This provides additional support to the conclusions drawn using the FM methodology. 32 Davidson and MacKinnon's (1981) Jtest can help assess the relative strength of trading volume and size in explaining the cross sectional differences in stock returns. Intuitively, Davidson and MacKinnon's (1981) Jtest for nonnested models works as follows. Suppose, Then, we want to compare two nonnested hypotheses, HI with we first find the predicted values of the dependent variable using (say) Next, we run HI using these predicted values as an additional explanatory variable and look for the significance of the coefficient on this additional variable. The coefficient of the additional variable will be significantly different from zero if the initial predicting equation, i.e., H2 is valid. We then repeat this procedure using predicted values from HI as an additional explanatory variable in H2 and evaluate significance of the coefficient. In our case, HI states that size explains returns on stocks while H2 states that trading volume explains the returns on stocks: HI: Rit i + b1u Ln (Size,,,) + ei, H2: R , + d1 Ln (Volume,i.) + h,, Given the two nonnested hypotheses, First, we carry out the Jtest as follows. we regress stock returns on trading volume according to H2 and find the predicted values for returns. Then, we run H using these predicted values of returns H2 is 1.71 (tvalue 2.13). Given that the coefficient on the additional variable (namely, returns predicted using H2) is significant, this half of the test provides support in favor of H2, i.e., trading volume explains stock returns. The second part of the test proceeds in reverse order. Namely, w< returns on size according to H 1 and find the predicted values for returns. e first regress Next, regress returns on these predicted values as an explanatory variable in addition to trading volume. We find that the average slope coefficient on trading volume is 0.80 (tvalue 0.72) while that on the additional variable is 1.05 (tvalue .60). Since the coefficient on the additional variable (namely, returns predicted using HI) is not significant, the conclusion is that the predicted values from H do not explain the returns significantly. Therefore, favor of we conclude that Davidson and MacKinnon's Jtest rejects size in trading volume (or liquidity) as the explanatory variable of the cross section of stock returns. Size or Volume?, Evidence from Test Portfolios The high crosssectional correlation between size and volume presents a problem in that the estimates lack precision in presence of multicollinearity. In this section we follow the methodology used, in a similar but different context, by Jegadeesh (1992); and construct test portfolios that have a low crosssectional returns. In contrast, controlling for size. volume has a strong negative relationship with returns even after Volume absorbs the role of size in explaining returns and this is true even when we include January months in the test. Test Portfolios Four types of test portfolios are examined to study the influence of size visa vis volume on the crosssectional returns. These portfolios are called size based, volume based, sizevolume based and volumesize based portfolios. The first type of test portfolios, called the 'size based' portfolios is constructed as follows: Once every year in July, the securities listed on the New York Stock Exchange (NYSE) are ranked based on the market capitalization of equity in the prior month (or June) and 20 sizebased groups are formed. Portfolio equally weighted represents an holding of the smallest 5 percent of the firms and portfolio 20 consists of an equally weighted portfolio of the largest 5 percent of the firms. This procedure is repeated once every year from 1963 to 1990. Appendix A, Panel A summarizes the time series averages of properties of this type of portfolios. The 'volume based' portfolios are formed in a similar manner except that the ranking of firms is done on the basis of average monthly trading volume over the last year (at least past six months). Appendix A, Panel B. Table I, The properties of these portfolios are summarized in II shows that the average correlation between size and urlnlimi in r thfaca nrnrtrf'nline ;e h;hlh /(0kr.t (" QC t, A / CX, formed as follows: every year in July, we form 10 size based groups as before. Each of these 10 size based groups is further partitioned into 3 volume based groups. breakpoints for volume based ranks are found each year by ranking all the NYSE listed securities into three equal groups on the basis of the average volume of trade over the prior year. This way, thirty 'sizevolume' based portfolios are formed once every year by using an equally weighted combination of member firms in each group. For example, thirty 'sizevolume' based portfolios for the year formed as follows: 1963 are 1963 we rank all the NYSE listed firms into ten groups on the basis of size at the end of the prior month (or June 30' of these ten groups is further subdivided into 3 volume based groups. 1963). Each one The breakpoints for volume are calculated by ranking all the NYSE listed groups into 3 equal groups on the basis of the average volume of trade over the prior year (beginning on July '1962 and ending on June 30' 1963). Thirty equally weighted portfolios are formed using these sizevolume based groups. year up to and including July 1' 1990 This procedure is repeated in July of every . Appendix A, Panel C shows the time series averages of properties of these portfolios. Finally, we form a fourth type of portfolios called the 'volumesize' portfolios. These are constructed by using a similar procedure as above except, now we first form 10 groups based on volume and then divide each of these groups into 3 sizebased groups. Appendix A, Panel D shows the properties of these portfolios. FamaMacBeth Rezressions The relationship between returns and explanatory variables, size and volume, is examined using the procedure developed by Fama and MacBeth (FM) (1973). A similar procedure is used by Fama and French (FF) (1992). Lys and Sabino (1992) point out that grouping based tests, of differences of means, may lack power. We do not use the usual means tests but instead use the regression approach that may have more power per Lys and Sabino (1992). Four sets of FM regressions are performed; one regression set for each type of portfolios. The FM regressions are carried out as follows: starting with July 1963, calculate log of size (LSZ,,,) and log of average volume (LVOL1,.i) for each portfolio. LSZ,,1, and LVOL,11 represent the June). respective values in the prior month (or The CRSP returns (expressed as a percentage change in the value of unit investment) are calculated over a period from July of year t through June of year t+1. The next observation is made in July of the following year (or year t+ 1). This provides a maximum of 28 observations per portfolio and the periods covered are nonoverlapping. The returns are regressed crosssectionally, each year on the explanatory variables. Table 1V shows the time series average of slopes of crosssectional yearly regressions of portfolio returns on size and volume. The average slopes should be 37 The slope coefficients on size or volume (individually) are significant in the size portfolios and in the volume portfolios. For example, as shown in Table IV Regression set 2 (volume based portfolios), the average slope coefficients (tstatistics) on size and volume are 1.83(2.51) and 1.61(2.68) respectively in the univariate regressions. The high correlation between size and volume in these portfolios (0.85 to 0.95 from table III) makes it impossible to decide whether the returns are related to size or to volume (because size and volume are very good proxies for each other). Additional tests are helpful in isolating the effect of size and volume on returns. This is achieved by using the sizevolume and volumesize portfolios; when we use sizevolume portfolios or volumesize portfolios, we are able to reduce the correlation between size and volume to about 0.35, as can be seen from table III. These sets of portfolios with reduced correlation (between size and volume) allow us to clarify the role of size visavis volume in explaining the variation in cross sectional returns. Table IV shows that the average slopes of FM regressions are stronger for volume relative to size, this is true regardless of whether we use only size (or volume) as an explanatory variable or use size and volume simultaneously. For example, as shown in Table VIRegression set 3 (sizevolume portfolios), the average slope coefficients (tvalues) on size and volume are 0.36(0.37) and 1.60( 2.03) respectively in the univariate regressions. Results shown in Table IVRegression set 4 (volumesize portfolios) are similar to the results in Regression set 3. These tests crosssectional variation in returns. Table V shows essentially similar results using a different proxy for liquidity (number of outstanding shares). In conclusion, size (independent of proxies for liquidity) is not related to crosssectional returns; any observed relation between returns and size may be due to the relation between size and liquidity; in contrast, there is a strong negative relationship between returns and liquidity (independent of size). Table VI shows that size has no relationship with returns in nonJanuary months but trading volume is negatively related to returns. set 3 (table VI), For example, in regression volume has a slope of 1.92 with a tvalue of 2.67 but size has no explanatory power, size has a slope with a wrong sign and a tvalue of 1.57 Table VII shows essentially similar results using a different proxy for liquidity (number of outstanding shares). Overall, the evidence suggests that liquidity proxies (trading volume or number of outstanding shares) may absorb the role of size in explaining the crosssection of average returns. Further, the liquidity effect is pervasive (not limited to January). Analysis of residuals In this section we use the methodology that was used by Chen (1983) to examine the explanatory power of CAPM relative to APT. The main idea is simple: we want to first allow size to explain the returns and then examine the residuals to see exercise by reversing the order and check f size can explain anything that liquidity may have missed. The basic approach is to carry out FM type crosssectional regressions as follows: Model t Ln sizee. .) + + d, Ln (volume.,.) Model + g, Ln (volume., ) + using monthly data for returns, size (m,). + m11, Ln (sizei,.1) we have 348 estimates of slopes on volume (d) and The time series average of these slopes and the associated standard errors are shown in table VIII. Similar analysis is done using two different horizons for returns, sixmonthly and yearly. In addition the entire analysis is repeated in non January months as well At first glance, this approach appears superfluous (see footnote #8) in that the multiple regression in Table , using both size and volume, seems do the same analysis. However , this approach different from the multiple regression analysis. Here, we hypothesize that we have two alternative models. model uses size as an explanatory variable and the second mode uses volume as an explanatory variable. The main objective, in this approach adequacy of a single model; i.e. we want to is to examine the see if one model leaves any information 40 In nonJanuary months, size has no additional explanatory power after allowing liquidity (volume) to explain the returns. For example, in monthly analysis, size has a slope of 0.04 and a tvalue of 0.73. Sixmonthly and yearly analysis shows similar results. Liquidity (volume) has significant explanatory power and the t values on liquidity are stronger than size, in the entire analysis. As pointed out by Chen (1983), the results in this section can be misleading if a variable is not significant, by itself as in table I, but attains significance in residual analysis. We do not have this type of a problem in our analysis. Overall, the evidence corroborates the suggestion in prior sections that size has no explanatory power in nonJanuary months but liquidity does explain returns throughout the year. These findings are robust to returns measured over different horizons (monthly, sixmonthly or yearly). Summary In Chapter than more liquid asse premia? In this chapt it was suggested that illiquid assets may provide higher returns ts. Is there any empirical support to this notion of liquidity er, we examine this question by analyzing the returns that are realized by stocks listed on the New York and American Stock exchanges. The results support the notion of liquidity premia. Our main result is that trading volume has a strong negative relationship with 41 control for the effect of size to distinguish between the role of size and trading volume. The relationship, between returns and trading volume, persists even after controlling for the role of size. This evidence strongly supports the notion of liquidity premia in financial markets, to the extent that trading volume is a good proxy for liquidity. Further, when we control for volume, the size does not explain any variation in stock returns. Another liquidity proxy like the number of outstanding shares, has a similar influence on the relation between size and stock returns. Simply stated, liquidity proxies absorb the role of size in explaining the crosssection of returns. These results are robust to different measurement horizons (for example, monthly returns, sixmonthly or yearly returns). Moreover, the relationship between liquidity and returns is not limited to January whereas sizereturn relationship is limited to January. The evidence suggests that liquidity provides a better description of the cross section of returns than size does. Specifically, liquidity may absorb the role of size and the liquidity effect is pervasive (not limited to January). Moreover, to the extent that size plays a major role in describing stock returns [Fama and French (1992)], it seems that liquidity premia may have a major role in the determination of relative asset prices. 42 TABLE AVERAGE SLOPES OF MONTHLY RETURNS ON BETA, SIZE, VOLU SHARES OUTSTANDING: JANUARY CROSSSECTIONAL REGRESSIONS OF ME OF TRADE AND THE NUMBER OF 963 TO DECEMBER The associated tstatistics are in parentheses Starting with January 1963, log of size (LSZn, ,_ log of volume (LVOL1.,), log of shares outstanding (LSHi,.) in the prior month and beta (using last 60 to 30 months of data) are calculated for each firm. These explanatory variables are matched with CRSP returns (expressed as percentage change in value of unit investment) over the current month. The next observation is made in the following month. provides a maximum of 348 observations per firm. Returns are regressed each month on the expj)lanatory variables. Average slopes are tlhe time series , I Beta LSZ LVOL LSH 0.06 (0.31) 0.14 (2.58) 0. 12 (2.95) 0.16 (3.13) 0.04 (0.21) 0.14 (2.68) 0.12 (1.30) 0.02 (0.25) .17 (1.33) 0.05 (0.38) 0.037 (0.25) 0.24 (1.81) 0.94 (3.37) 0.88 (2.70) I 43 TABLE II AVERAGE CORRELATIONS BETWEEN SIZE VOLUME OF TRADE AND THE NUMBER OF SHARES OUTSTANDING: JANUARY 1963 TO DECEMBER All correlations are significant at 0.01 (0.0001) level Starting with January 1963, log of size (LSZ .,,I), log of volume (LVOL,.,.), log of shares outstanding (LSH,,.) in the prior month are calculated for each firm. The next observation is made in the following month. This provides a maximum of 348 observations per firm. The correlations are calculated simply from the pooled time series and crosssectional data. LSZ LVOL LVOL 0.72 LSH 0.88 0.84 a *2 44 , 0 . *u  c "3a: 4 4 3o a ;f 0 Cu ^ 5 " 2 t . s VU  c  ca" *2 F 1 f C7,cc C J ' F: .2 'S .2^^o . '4) 13 ^^^ 6^ 2 " "i 0 Cso* C ^S 5 cn Cu 'S C 'S  >'' ft t4 fc. %  7 3> 'S '" .9 U h aod^  ^ #'' Cs^* 'g fT ; o o _ CU^ '^S L14 C.Q O ft.i ^ ____ S MDS . ^ 4 ti 3 < 13  0 .4' . CU . 'I ^ g* C.' O5 S Cu a^ ^' cU 2?ES*C?>" _n U 3 C 1 eII C0 ft3 oo 5 ^: 4 . C ^"c 0 C)A I.f I. o __  *0 2 c CU U G) ft c 5 o 0 S * Fu  cu ^ .s C3 * S CU F 0 .') .9U  *. z 3 E = t ^ In i S " Cu C) 3 0 Cu Se CU. v 2 CU C4. 3  CU TABLE AVERAGE SLOPES OF FAMAMACBETH (FM) REGRESSIONS, 6/63 TO 6/90 Regression Set 1 Size based Portfolios 20 portfolios per year for 28 years Coefficient of Size 1.56 ( .77 (0.50 h,. Coefficient of Volume 3.02 (2.14) 1.03 (0.54) Regression Set Volume based Portfolios 20 portfolios per year for years Coefficient of 1.83 (2 2.09 (0.9 b,, Coefficient of Volume 1.61 (2.68) .27 (0 ression Set 3 SizeVolume Portfolios 30 portfolios per year for 28 years b,, Coefficient of 0.36 (0 0.24 (0.22) Coefficient of Volume .60 ( .24 (1.28) Regression Set 4 VolumeSize Portfolios 30 portfolios and years bh, Coefficient of 1.05 ( 0.90 (0.81) b,, Coefficient of Volume (2.07) (1.23) The associated tstatistics are in parentheses Four separate sets of FM regressions are carried out as follows: starting with July calculate log of size (LSZ1.,) and log of average volume (LVOL,4,) for each portfolio. LS 1963, we LVOL1., represent the respective values in the prior month (or June). The CRSP returns (expressed as a percentage change in the value of unit investment) are calculated over a period from July of year t through June of year t+ 1. The next observation is made in July of the following year (or year t+ 1). This provides a maximum of 28 observations per portfolio and the periods covered are non overlapping. The returns are regressed crosssectionally, each year on the explanatory variables. The time series averages of slopes of crosssectional yearly regressions of portfolio returns on size and volume are presented below. The tvalues for average slopes are based on the time series standard error of the estimates of mean slopes, tvalues are shown in parentheses. For each of the four regression sets, the basic crosssectional regression is: 46 TABLE V AVERAGE SLOPES OF FAMAMACBETH (FM) REGRESSIONS, 6/63 TO 6/90 Regression Set 1 Size based Portfolios 20 portfolios per year for 28 years b,, Coefficient of 1.56 (1.91) 0.63 (0.29 b,, Coefficient of LSH 2.29 (1.95) 1.68 (0.62) Regression Set LSH based Portfolios 20 portfolios per year for 28 years Coefficient of Size 1.77 (2 2.04 (0.74 b,, Coefficient of LSH 2.15 (2 0.37 (0. session Set 3 SizeLSH Portfolios 30 portfolios per year for 2 years bh, Coefficient 0.88 0.77 0.47 b,, Coefficient of LSH 1.82 (2.98) 0.83 (0.54) Regression Set 4 LSHSize Portfolios 30 portfolios and years b,, Coefficient of 1.41 ( 0.88 (0.48) Coefficient of LSH 2.29 (3 . (3 1.63 The associated tstatistics are in parentheses Four separate sets of FM regressions are carried out as follows: starting with July 963, we calculate log of size (LSZ;,,) and log of average LSH (LSH,,.) for each portfolio. LSZi,,, and LSH,1,.1 represent the respective values in the prior month (or June). The CRSP returns (expressed as a percentage change in the value of unit investment) are calculated over a period from July of year t through June of year t + 1. The next observation is made in July of the following year (or year t+ I). This provides a maximum of 28 observations per portfolio and the periods covered are nonoverlapping. The returns are regressed crosssectionally, each year on the explanatory variables. The time series averages of slopes of crosssectional early regressions of portfolio returns on size and LSH are presented below. The tvalues for average slopes are based on the time series standard error of the estimates of mean slopes. tvalues are shown in parentheses. For each of the four regression sets, the basic crosssectional regression 47 TABLE VI EXCLUDING JANUARY, AVERAGE SLOPES OF FM REGRESSIONS, 6/63 TO 6/90 Regression Set Size based Portfolios 20 portfolios per year for 28 years b,, Coefficient 0.09 of Size (0.12) (1.20) b,, Coefficient of Volume 0.41 (0.34) 2.55 Regression Set Volume based Portfolios 20 portfolios per year for Regression Set 3 SizeVolume Portfolios 30 portfolios per year for 28 years years Coefficient of 1.09 (1.83) 0.49 (0.24 h,, Coefficient of b,, Coefficient of Volume 0.96 (1.94) 0.47 (0.27) Coefficient of Volume 1.92 (2.67) 2.12 (2.38) Regression Set 4 VolumeSize Portfolios 30 portfolios and years b,, Coefficient of Size (0.90) Coefficient of Volume 1.99 (3.06) 1.99 ( 2.58) The associated tstatistics are in parentheses. Four separate sets of FM regressions are carried out : starting with July 1963, we calculate log of size (LSZI,) and log of average volume (LVOL .,) for each portfolio. LSZAI and LVOL,,, represent the respective values in the prior month (or June). The CRSP returns (expressed as a percentage change in the value of unit investment) are calculated over a period from July of year t through June of year t+ 1; returns in January are ignored. The next observation is made in July of the following year (or year t+1). This provides a maximum of 28 observations per portfolio and the periods covered are nonoverlapping. The returns are regressed crosssectionally, each year on the explanatory variables. The time series averages of slopes of crosssectional yearly regressions of portfolio returns on size and volume are presented below. The tvalues for average slopes are based on the time series standard error of the estimates of mean slopes, tvalues are shown in parentheses. For each of the four regression sets, the basic crosssectional regression is: 48 TABLE VII EXCLUDING JANUARY, AVERAGE SLOPES OF FM REGRESSIONS, 6/63 TO 6/90 Regression Set Size based Portfolios 20 portfolios per year for 28 years b,, Coefficient of 0.09 (0.12) (1.49) b., Coefficient of LSH 0.18 (0.18) 5.31 (1 Regression Set LSH based Portfolios 20 portfolios per year for 28 years efficient of 0.72 (1.19) 0.79 (0.32) b_, Coefficient of LSH 0.87 1.85 (1.20) (0.65) Regression Set SizeLSH Portfolios 30 portfolios per year for 2 years b,, Coefficient of 1.24(1 b,, Coefficient of LSH 1.80 (2.68) 3.40 (2.42) Regression Set 4 LSHSize Portfolios 30 portfolios and years b,, Coefficient of 0.61 ( 0.60) b Coefficient of LSH 2.63 ( 4.10( 4.11) 2.94) The associated tstatistics are in parentheses Four separate sets of FM regressions are carried out: starting with July 1963, we calculate log of size (LSZ .,) and log of the number of shares outstanding (LSHi.i) for each portfolio. LSZ.,I and LSH 1., represent the respective values in the prior month (or June). CRSP returns (expressed as a percentage change in the value of unit investment) are calculated over a period from July of year t through June of year t+ 1; returns in January are ignored. The next observation is made in July of the following year (or year t+1). This provides a maximum of 28 observations per portfolio and the periods covered are nonoverlapping. The returns are regressed crosssectionally, each year on the explanatory variables. The time series averages of slopes of crosssectional yearly regressions of portfolio returns on size and volume are presented below. The tvalues for average slopes are based on the time series standard error of the slope estimates, tvalues are shown in parentheses. For each of the four regression sets, the basic crosssectional regression is: 49 TABLE VIII RESIDUAL ANALYSIS , AVERAGE SLOPES OF FM REGRESSIONS, 6/63 TO 6/90 Monthly All NoJan Months Regression Set 0.18 0.09 1 (4.34) (2.65) Volume, d SixMonthly All NoJan months 1.20 0.63 (4.45) (3.09) Yearly All NoJan Months 2.28 1.41 (3.59) (2.58) Regression Set 2 0.04 (0.73) 0.20 (0.56) 0.62 (0.77) The associated tstatistics are in parentheses. Using Chen's (1983) methodology, returns are regressed crosssectionally on size (volume) and the residuals from this regression are regressed on volume (size). This analysis is carried out in six different ways, using three different horizons for returns (monthly, sixmonthly and yearly returns) and using two different calendar periods for each horizon (including January and excluding January). The time series averages of slopes of crosssectional FM regressions are presented below. slopes show the average effect of each explanatory variable on the residual returns. The average The tvalues for average slopes are based on the time series standard error of the estimates of mean slopes, tvalues are shown in parentheses. For each of the regression sets, the basic crosssectional regression is: Regression Set 1:R,1 = a, + b, Ln (size, ) + = c1 + d1 Ln (volume..) + Regression Set g, Ln volumeme~. = h, + m, Ln (size,.,) + TlrP ctmr.a c snfl eruoe ,Vcli rtn nnnn. .l\t u. II( n k.*ln. 41th' *r. CHAPTER 3 IMPACT OF LIQUIDITY ON PREMIA/DISCOUNTS IN CLOSEDEND FUNDS Introduction Closedend funds are traded bundles of traded assets. The market values of these bundles are often different from the collective market values of their contents, leading to observed premia or discounts. At first glance, these premia and discounts appear irrational because they seem to suggest that claims to identical cash flows may have different prices. Several explanations of discounts have been proposed in extant studies'. Most of these explanations motivate discounts through some special costs associated with the bundling of assets. However, these studies do not make any particular reference to the form of the bundle (openend or closedend) or the type of bundled assets (equity or bond). As a result most of the existing hypotheses are unable to explain the following observations: Premia and discounts may exist simultaneously across funds (Barclay et 1993). Different types of funds (equity or bond) may have different premia/discounts on average (Barclay et al.,1993). 51 Closedend funds trade at zero discounts when they are openended (by legal fiat)2. Liquidity considerations, potentially explain the above observations and thus provide an appealing explanation of the premia/discount phenomenon. The connection between premia/discounts and liquidity is straightforward. Amihud and Mendelson (1986) suggest that in equilibrium, the expected friction in the trading process is reflected in current market value. This implies that a bundle of assets is worth less than its contents if it is less liquid than its contents In the context of closedend funds: discounts are observed when a fund is less liquid than the assets in its portfolio and, premia are likely when a fund is more liquid than the holdings in its portfolio. Subrahmanyam (1992) and Gorton and Pennacchi (1992) provide an explanation for why the liquidity of a fund may be different from the liquidity of its portfolio. They suggest that security specific private information is diversified away in a 'traded basket of securities' (fund). This information diversification has, potentially, two conflicting effects on liquidity of a traded basket: first, the uninformed investors face less information asymmetry (when trading a basket) and this may improve liquidity. Second, the basket has lower trading activity (than the assets in the basket) 2 This is documented by Brickley and Schallheim (1985). Fund related costs that potentially justify discounts (management fees, taxtiming loss etc.) are applicable to openend funds as well. The 'Cost' based explanations of discounts suggest that the market value of an openend fund overstates its true value. This implies that, in equilibrium, the openend funds should be fully ^ ^ I ^ > ^ ^ L  ^  ^ ^  > < 52 because the informed traders are driven away and the per share cost of trading may go up to the extent that fixed costs of market making are spread over a smaller trading base4. The net impact of these two conflicting effects on liquidity may increase or decrease the liquidity of a fund relative to its portfolio and this, in turn, results in observed premia or discounts. The objective of this chapter is to examine the relationship between liquidity and premia/discounts. We document the influence of liquidity on premia/discounts using a variety of techniques including a single latent variable specification. results strongly support the liquidity conjecture: funds with higher liquidity, as measured by proxies for trading activity, have higher premia (or lower discounts) than funds with lower liquidity. This relationship is observed in equity funds as well as in bond funds. Further, the results are robust to various assumptions about model parameters and error structures. The organization of this chapter is as follows: The first section develops testable hypotheses relating liquidity to premia/discounts. The second section presents data and methodology. Results are discussed in the third section. The fourth section presents sensitivity analyses. The fifth section shows informal evidence suggesting that closedend funds may behave like lowvolume stocks. The final section concludes. 53 Liquidity and Premia/Discounts How is Liquidity Related to Premia/Discounts? As discussed Chapter 1, the impact of liquidity on market value may be easily stated, namely the market value of a traded asset is equal to the 'fundamental value' minus the expected loss due to illiquidity. This simple notion may explain the premia/discounts in closedend funds. For the sake of exposition let's assume that a closedend fund holds a fixed portfolio of traded assets, although in reality the portfolio may be dynamic. An investor can buy a claim (to a given set of cash flows) directly in the market or indirectly through a fund. The claim ownership through a fund would be worth more/less than direct ownership if it (ownership through a fund) reduces/increases expected costs of trading. For example, consider a traded security (like I.B.M.) and suppose that one share of such a security represents a claim to cash flows that would be worth $10 if there were no trading costs. expected trading costs were $1 However, if the (value of liquidity) then the market value of such a share would be $9 ( or $10$1). Now consider a closedend fund that holds one such share as the only asset in its 'portfolio' and the fund issues one share. The net asset value (NAV) of the fund would be $9 (the market value of fund's assets). of the fund represents a claim to cash flows that are worth $ One share 0 if the fund share can hP tradC d x/ith 7,=rr tr..rlno r',tv ;ftL//r f"tlh trrl,'^n rnctc 1E, r, 4j) th1n thn, market value of the fund share would be $8 (or $10$2). In this situation the market value of the fund, $8, would be less than the NAV of the fund, $9, and this results in a discount of $1 on the fund share relative to the NAV of the fund. Conversely, if the trading costs of the fund share were $0.5, then the market value of the fund would be $9.50 and the fund would trade at a premium of $ 0.50 relative to its NAV Notice that the fund would be at a premium/discount depending on whether it is more liquid or less liquid than its portfolio.6 Testable Hypotheses Our central conjecture is that premia/discounts vary depending on the liquidity of the fund shares relative to the liquidity of the assets held in the fund's portfolio. A direct test of this notion is difficult to implement without any information about the composition of a fund's portfolio. However, an indirect test can be devised using the classical way of analyzing variability of an endogenous attribute by examining any variation 'withingroups' and 'acrossgroups' This approach results in two hypotheses described below: Within a group of funds that hold similar assets (equity or bond), premia increase (discounts decrease) as the liquidity of iunds ' shares " Note that the openend funds (OEFs) are traded in essentially a batch market. Investors place buy and sell orders directly with the fund and the fund can cross orders at the end of the A9I ; \/at thi nraf,>\ increases. To see this, let us consider two closedend funds that hold identical assets but have different liquidity . The NAVs of both the funds would be equal to each other (by design) but the market values would be different depending on liquidity. A fund that is more liquid would have a higher market value resulting in a higher premium or lower discount than the other fund. This is illustrated in more detail in Appendix B. 2) Average premia/discounts may be different across different types of funds (equity versus bond). Let us consider two groups of funds, equity funds and bond funds. The average trading costs for equity assets are likely to be different from bond assets because these costs relate to different types of assets that are traded in different trading structures. On the other hand, the average trading costs for the shares of these funds are likely to be similar because they both are equity securities that are traded in similar trading environments. main distinction between the two groups of funds is in respect to the portfolio assets and differential liquidity of underlying portfolioassets may result in different average premia/discounts across different types of funds. Appendix B provides a more detailed illustration of this idea. 56 Proxies for Liquidity Literature on liquidity suggests that trading costs decrease in trading activity and increase in the volatility of a traded asset.8 We use weekly volume of trade, weekly dollar volume of trade and percentage of shares traded in a week as proxies for trading activity. Normalized price range (maximum price minus minimum price divided by minimum price) over a week is used as a proxy for volatility. Merton (1987) proposes that liquidity increases in asset. 'investor recognition' for the This suggests that the number of outstanding shares may be a proxy for liquidity. Stoll and Whaley (1983) suggest that trading costs decrease in firm size. This suggests that market capitalization (size) may be a good measure of liquidity. The next section describes the data and methodology that we use to examine the hypotheses outlined in this section. Data and Methodology Data Weekly data for funds listed on the NYSE is obtained from the CRSP (Center for Research of Securities Prices) daily tapes. Each week covers all the trading days between Friday and Thursday. 19/1l/O1 The net asset value (NAV) data from iQC5 opnrrnaic1t\ nrrnl\irl\f hlp\ flr lcormnnt A drli/icnrc 1/4/88 to ThP N1AV calculated upon close of trading on the Wall Street Journal. Thursday and are reported in the Friday issue of The sample contains 18 domestic equity funds and 90 bond funds.9 The natural logarithm of all raw variables is used to reduce the skewness in the data; this way the outliers are 'pulled in' and do not overwhelm the analysis. Log of firm size (LSZ,) is defined as the natural logarithm of total market capitalization at the end of week t, for a given fund i. Log of volume of trade (LVOLDi) is defined as the natural logarithm of the volume of shares traded in week t. LDOL, is the log of dollar volume of trade in week t, for fund i. The log of outstanding shares (LSHH,) is defined as the natural logarithm of the number of shares outstanding at the end of week t. LTURN, is log of percentage of outstanding shares traded in week t. Premium is measured as prices NAV1i NAVi, X 100. LPREM., is defined as price, Log( NV). NAV, LSIG1, is a measure of volatility defined as the log of normalized price range (maximum trade price minus minimum trade price divided by minimum trade price) for fund i over week t. 58 Methodology We use the ordinary least square (OLS) method as well as the latent variable method to examine the relationship between premia/discounts and proxies for liquidity. The OLS specification is: LPREM,, + 3, (liquidity proxy),1,_l where, i is an index for funds and t is an index for time. The OLS method is known to be biased when explanatory variables are measured with error. In the context of liquidity, the proxies for liquidity are likely to measure the true liquidity with considerable error; and OLS may be unreliable for the purpose of estimation. In such situations, the latent variable approach has many advantages over the OLS method and this is discussed next. The Latent Variable specification treats liquidity as an variable."~ Several measurable variables such as size (LSZ1,,), 'unobserved' volume of trade (LVOL1,t.), dollar volume of trade (LDOL1.[.) and turnover rate (LTURN,L.I) are treated as errorprone proxies for liquidity and the attendant errorsinvariables are explicitly specified. The latent variable model is: LPREM1, = ^ ( 4) + Oi,, = 21 (4i) + 02it = 1, =  3 (ki) + 3it = P4, (Pi) + 4 where, Y3 and Y4... are some proxies that measure unobservable liquidity (4) with error (0), the subscripts explain the timeseries and crosssectional restrictions that are imposed on the parameters. system if we have more proxies available for <. We can add more equations to the These equations are estimated simultaneously as a system with f, 4 and 0 as parameters. The econometric setup implies restrictions on the covariances (among observed variables), and these restrictions can be expressed in terms of the parameters of the system. Using this underlying principle, the estimation procedure uses information in the sample covariance structure to estimate parameter values." Such a technique provides consistent estimates of parameters. Further, any omitted variables in the list of 'Y' variables above, do not affect the consistency of parameter estimates.'2 Discussion of results Table IX shows descriptive statistics of the sample. The mean premia are about 12% for equity funds and about 1% for bond funds. Several tests were carried out to examine if these mean premia are statistically same or different, and all tests indicate that the mean premia are different at 0.0 level. For example, the equality 60 of means test (ttest) for means from two independent random samples [See, Greene (1990)] was carried out. The null hypothesis that the mean premia are same for bond funds and equity funds is rejected with a tvalue of 55.64 (for the test statistic) under assumption of unequal variances across samples; and under the assumption equal variances, the tvalue is 66.89. Such rejections may be misleading if the underlying distributions (of means) are not approximately normal.'3 To ensure that our results are not driven by the violation of distributional assumptions, non parametric tests [See, Conover (1980)] were also carried out. The Wilcoxon ranksum test rejects the hypothesis of equal premia (across groups of bond and equity funds) with a Z score of 52.33, which is significant at 0.01 The KruskalWallis test rejects the null with chisquare (1 degree of freedom) value of 2738.6, value of 0.0001. which has a p Thus, hypothesis #2 is supported in our sample; average premia in equity funds are significantly different from average premia in bond funds. Table X examines equity funds. There is a significant crosssectional relationship between premia/discounts and various liquidity proxies. For example, estimates of slopes for all the variables are significant at level. This finding supports Hypothesis #1 in the first section; premia increase as funds' liquidity increases given that the underlying assets have similar liquidity. Literature suggests that the cost of market making increase as volatility increases." Thus, we would * I I *x j j ,^ r j * ^ .*^* r.n *r\k t ..< ^ E... .L  .. _.I / .^ j expect liquidity to decrease as volatility (LSIG) increases. Table X shows that the slope on volatility (LSIG) is negative, supporting the notion that premia are positively related to liquidity. Table Xl shows similar results for bond funds. The ordinary least square method may be subject to errorsinvariables bias. Such a bias attenuates the slope towards zero and the effect on tvalue is ambiguous. We use latent variable technique to correct for errorsinvariables later on. Further, the OLS method, as used in Tables X and XI, may be biased to the extent log of price may, indirectly, introduce a lagged dependent variable bias in small samples. account for this potential bias, we carried out crosssectional regressions on a weekly basis; and the result of this analysis was substantively similar. Table XII shows the results of latent variable analysis using the data on equity funds. Each row in the table represents a separate model. We would expect positive coefficients on liquidity for all variables except for LSIG. All models support the notion that premia are increasing in liquidity (Hypothesis #1 in the first section). For example, the slopes on liquidity (Ct) with premia (LPREM) are positive and significant level. The slopes on liquidity (4) for various proxies for liquidity, are in the expected direction (except for LSIG in a few models) and significant at level. These results, taken together, suggest a positive relationship between premia and liquidity. Bond funds show essentially similar behavior as shown in Table V , adding further support to the liquidity conjecture. 62 significantly different across different types of funds. For example, discounts are more likely in equity funds than in bond funds. This finding suggests that the observed mean differences in premia (as shown in Table IX) are not driven by outliers. To examine the stability of findings, different subperiods. we repeated the above analyses in several The findings (not reported here) are substantively similar and lend further credence to the above discussion. Overall, in our data, we find strong support for the notion that premia/discounts in closedend funds are related to proxies for liquidity of the claims issued by the funds. In the next section we examine the robustness of these results to various econometric specifications. Sensitivity Analysis The sensitivity of the results described above is examined with respect to various assumptions about the intercepts, coefficients and the error structures. Specifically, we examine 1) panel data models with fixed and random effects in the intercepts, 2) random coefficient model, and 3) Seemingly unrelated regression (SUR) models. All of these methods are discussed in Greene (1990). The results are generally found to be robust to various specifications and thus provide additional support to the findings described above. 63 This model assumes that differences across time periods (regarding average premia and other industry wide effects) are reflected in the intercept. The model is written as: LPREMI, + / (liquidity proxy)1,, is an index for funds and t is an index for time. The slopes on liquidity proxies using the FEM are reported in coefficients of all the explanatory variables are significant at Table XV level and the signs of the coefficients are exactly in line with the implications of the liquidity conjecture. For example the turnover rate (LTURN) has a coefficient of 0.023 and a highly significant tvalue of 9.30. The volatility measure (LSIG) is negatively related premia with a coefficient of 0.036 with an associated tvalue of 1 2) Panel Data Analysis: Random Effect Model (REM) The REM specification allows the intercepts to be random across time in contrast to the earlier FEM that treats the variation in intercepts as parametric shifts. The REM is more general in that the selected time periods are treated as a sample from a broader population. The REM is written as: LPREM, where j3 (liquidity proxy).tl is an index for funds, t is an index for time and u, is a meanzero random disturbance reflecting a random industry wide changes in the level of average premia and other effects common to the industry. = Ot, = + 64 3) Random Coefficients Model (RCM) The RCM is even more general than the REM in that the coefficients, including the intercept, are treated as random across time while the crosscorrelation and heteroscedasticity of the coefficients is taken into account. The RCM specification does not assume that the parameter estimates are independent drawings from a stationary distribution. Therefore, the RCM is more general than the classical Fama and MacBeth approach to aggregating a time series of crosssectional estimates. RCM may be stated as: LPREM, = at. + / (liquidity proxy),.l + error, + error, where i is an index for funds, t is an index for time. Variation in the coefficients reflect industry wide random change in the level of premia as well as the changes in the impact of liquidity on premia. Table XV reports the results of the RCM analysis. All the coefficients on liquidity proxies continue to be significant at level. 4) Seemingly Unrelated Regression Models (SURI and SUR2) The SUR models allow crossequation correlation, SUR models. we examine two separate The basic model is specified as a following system of equations: LPREMi, + 3 (liquidity proxy);i,, + E.i 65 SURI accounts for the effects of omitted variables that may be common across funds. SUR2 admits the notion that repeated observations on the same entity may produce errors that are correlated across time. The slopes and associated significance are reported in Table XV The results on SUR2 may not be reliable because we were constrained to using no more than ten weeks of data in SUR2 analysis (this limitation is due to identification). The results of SURI (and most of SUR2) are substantively similar to the results seen earlier. In summary, Table XV provides overwhelming evidence that the observed relationship between liquidity and premia is unlikely to be an artifact of assumptions in our model specification. Some informal evidence Are the Returns on Closedend Funds Related to LowVolume Stocks? Lee et al. (1991) examine the relationship between premia and the return behavior of the smallest decile of stocks on the NYSE. Chen, Kan and Miller (1993) take a second look at such a relationship and conclude that there is no special relationship between premia and the small stocks. In similar spirit, we explore the whether the returns realized by closedend funds are related to thinly traded (lowest trading volume decile) securities in any special way. An existence of such a rel2tinnvhin xvnnld cinooa ct th'sn tha nlvr'i'l:,nir f'nnrte ih,n o h~uoa Ii1 illirhi1irl 66 related to highlytraded securities.is Such evidence would be supportive of the conjecture that liquidity of a fund may be a major determinant of the market value and, therefore, may be related to premia in closedend funds. We tested the following timeseries specification: + i Xi, where t is an index for months spanning January 1963 to December 1991 and is an index for volumedecile portfolios of NYSE listed stocks. Y, is the excess return realized by the closedend fund industry in month t. The excess return was defined as the valueweighted return realized by the closedend industry minus the return on the value weighted market. The stocks listed on NYSE (excluding closedend funds) were divided into ten portfolios based on their volume of trade in the prior month. X, denotes valueweighted returns realized by the i1' decile in excess of the return realized on the valueweighted market, in month t. Table XVI shows that the lowest volume decile has the strongest relationship with the excess returns realized by the closedend industry. The R2 for the lowest decile is about 34% compared to for the smallest size decile and 0.33 for the largest volume decile. Further, the R2 declines almost monotonically as we look at higher deciles. Inclusion of dummies for January did not substantively affect the results reported in Table XVI. Overall, the results in Table XVI strongly suggest that returns on closedend funds may comove with returns on lowvolume securities. This = a; 67 result is surprising because, as mentioned above, the fundamentals of closedend funds are related to highvolume securities and not as much to lowvolume securities. Thinness of trading, despite the fundamentals, seems to affect the return behavior of closedend funds; and to this extent the results are consistent with the notion that market values of closedend funds, and therefore premia, may be related to liquidity. Summary We suggest that illiquidity or expected frictions in the trading process, may influence premia/discounts observed in closedend funds. Specifically, we conjecture that premia (discounts) are observed when claims issued by the fund are more (less) liquid than the underlying assets. The liquidity hypothesis is appealing, because it extends extant literature in several important ways. In particular, it is consistent with premia as well as discounts, it predicts that average premia/discounts may be different for different types of funds (equity versus bond) and it explicitly recognizes the trading frictions that may restrict apparent arbitrage opportunities in presence of premia/discounts. Our evidence strongly supports the liquidity conjecture. For example, funds with higher liquidity, as measured by proxies for trading activity and volatility, have higher premia (or lower discounts) than funds with lower liquidity. is observed in equity funds as well as in bond funds. This relationship Consistent with the liquidity 68 than in bond funds. Further, the observed relation between liquidity and premia is robust to various assumptions about parameters and error structures. Examination of excess returns suggests that closedend funds behave like illiquid securities, although the funds hold highly liquid securities in their portfolios. Overall, the liquidity hypothesis is supported in our data and may offer a richer description of the premia/discount phenomenon in conjunction with existing hypotheses. 69 TABLE [X DESCRIPTIVE STATISTICS OF THE SAMPLE. CLOSEDEND EQUITY JANUARY AND BOND FUNDS LISTED ON THE NYSE TO DECEMBER Fnnn January 4, 1988, log of (LDOL1,), size (LSZ,,), log of volume (LVOL,,), log of dollar volume of trade log of shares outstanding (LSH,,), loh of turnover rate (LTURN,,), lo of volatility (LSIG,,) and log of premium (LPREM,,) are calculated or each fund mon a weekly basis for about 208 weeks. Means and standard deviations, in parentheses, of variables are r:cpoted above. Equity Funds Bond Funds LPREMI 0.12 (0.10) 0.01 (0.07) LSZ, 19.07 (1.25) 18.77 (0.95) LVOL, 11.00 (1.58) 11.31 (1.56) LDOLi, 13.64 (1.46) 13.65 (1.37) LSHI 16.44 (1.25) 16.44 (1.07) LSIG, 3.85 (0.65) 3.86 (0.56) LTURN,, 0.83 (0.79) 0.52 (0.74) NUMBER OF FUNDS 18 90 TABLE AVERAGE SLOPES OF CROSSSECTIONAL REGRESSIONS. CLOSEDEND EQUITY FUNDS LISTED ON THE NYSE: JANUARY TO DECEMBER 1991 From January 4 1988, log of size (LSZ,,), (LVOL of dollar volume trade (LDOL,,), log of shares outlsiand ing (LSH,,), log of turnover rate (LTURN,), log volatility (LSIG,,) and log of premium (LPREM,) are calculated for" each fund on a weekly basis. The sample is divided into twenty periods of about 10 weeks each (partial pooling). Premia/di SCmUntl.S, expressed as LPREM,,, are regressed on the explanatory variables in each of the twenty periods. This provides 20 crcsssectiConal regressions on the semipoo)led sample, or one for each period. Using the classical FamaMacBeth appr )ach, the average sh lpsc are the timese ines averages o, the slopes obtained from the 20 regressions. The Lstatistic is simply slope divided by its timeseries standard across the 20 regressions. basic crosssectional LPREM,, irssion + h, X,, LVOL,.,1 LDOL., 1 LSZ,., LSH., LTURN,., I LSIG,., 0.008 (7.18) 0.02 (15.77) 0.022 (10.73) 0.006 (4.02) 0.019 (4.03) 0.028(7.48) error loh, of volume 71 TABLE XI AVERAGE SLOPES OF CROSSSECTIONAL REGRESSIONS. CLOSEDEND BOND FUND JANUARY LISTED ON THE NYSE 1988 TO DECEMBER From January 4, 1988, log size (LSZ,,), lor of volume (LVOL ,), log of dollar volume of trade (LDOL,), log of shares outstanding (LSH,1), log of turnover rate (LTURN,,), log of volatility (LSIG,,) and log premium (LPREM,,) are calculated for each fund on a weekly basis. The sample is divided into twenty periods of about 10 weeks each (partial pooling). Premia/discounts, expressedC as LPREM,,, are regressed on the explanatory variables each period. This provides 20 estimates from the semipooled sample, one for each period. Using the classical FamaMacbeth approach, the average are the time series averages of the slopes obtained from the 20 regressions. The (statistic is simply the average slope divided by its time series standard error ac ross the 20 regressions. The basic crosssectional LPREM,, regression is: + h, X, LVOL,.,, LDOL,1 LSZ., LSH,,, LTURN..II LSIG,.1 0.007 (3.01) 0.011 (5.05) 0.022 (1 1.31) 0.014 (5.21) 0.000 (0.040) 0.015(3.98) TABLE XII LATENT VARIABLE MODELS AVERAGE SLOPES OF CROSSSECTIONAL ANALYSES. CLOSEDEND EQUITY FUND JANUARY 1988 TO DECENT LISTED ON NYSE: 4BER 1991 From January 4, 1988. log of size (L SZi), og of volume (LVOL1i), og of dollar volume of trade (LDOLI,), log iof shares outstanding LSHI,), log of turnover rate (LTURN;,), of volatilit (LSIG,) and lo of premium LPREM1,) are calculated for each fund on a weekly basis. The sam of about 10 weeks each. A ple is divided into twenty periods system of equations is estimated every period provides 20 estimates from the semipooled samp one for each period. Using the classical FamaMacbeth approach, the sl the average slopes opes obtained from the 20 regressions. are the tstatistic timeseries averages o is simply the average slope divided by its timeseries standard error across the 20 regressions. The basic stem of equations is: LPREM, = P, 41) + 0,1, 4i) + ,,, (4.) + 0 j, P= 34l ( where, i=. to 18 equity funds and t= to 20 periods. 2. Y3 and Y4 are some proxies that measure unobservah uidity (4) with error (6). These equations are estimated simultaneously as a sy stem with ) and 0 as parameters estimates are consistent [See Bollen 1989) for details liquidity proxies are TABLE XII (CONTINUED) LATENT VARIABLE MODELS AVERAGE SLOPES OF CROSS SECTIONAL ANALYSES. CLOSEDEND EQUITY FUNDS LISTED ON NYSE: JANUARY 1988 TO DECEMBER Each row in the above shows estimates karate mod of simultaneous equations, and each column (within a given row cell entries represent estimates of coeffi is a separate equation within the model. clients (or estimates of /3) and tvalues of estimates are shown in parentheses LPREM LVOL LDOL LSZ LSH LTURN LSIG 0.023 (9.77) 0.99 (12.30) 1.52 (12. I1) 0.014 (6.83) 1.89 (9.37) 1.08 (8.10) 0.023 (7.76) 1.17 (7.86) 0.58 (12.12) 0.031 (12.42) 1.50 (19.08) 1.05 (26.57) 0.022 (9.21) 2.40 (11.51) 0.68 (13.23) 0.040 (11.93) 1.36 (7.00) 0.35 (10.18) 0.0.090 0.72 (4.14) 0.15 (6.23) 0.083 (5.98) 0.57 (3.83) 0.13 (1.77) 0.062 (5.45) 0.75 (5.48) 0.23 (7.13) 0.026 (4.64) 0.51 (3.90) 0.26 (0.87) 0.022(6.49) 1.14 (4.88) 1.54 (9.26) 0.052(2.88) 0.010 (2.91) 1.92 (8.84) 1.03 (7.31) 0.05 (2.96) 0.011 (2.67) 0.83 (5.42) 0.93 (9.80) 0.14 (7.01) 0.029 (7.84) 1.55 (10.93) 1.06 (12.57) 0.041(1.98) 0.022 (9.60) 2.14 (16.98) 0.73 (18.56) 0.001(0.12) 0.055 (4.48) 0.77 (5.27) 0.48 (2.87) 0.044(1.09) 0.079 (3.78) 0.50 (5.22) 0.07 (1.02) 0.27(4.72) 0.006 (0.53) 0.20 (2.91) 0.70 (2.95) 0.05 (0.49) 74 TABLE XIII LATENT VARIABLE MODELS. AVERAGE SLOPES OF CROSSSECTIONAL ANALY SES. CLOSEDEND BOND FUNDS LISTED ON NYSE: JANUARY 1988 TO DECEMBER 199 From January 4, 1988, log of volume of trade (LDOL;J, size (LSZ;,), lo log of shares outstanding (LSH;,), 'olume (LVOLt), log of dollar log of turnover rate (LTURN1), log of volatility (LSIG,,) and log of premium (LPREM,,) are calculated for each fund on a weekly basis. The sample is divided into twenty periods of about 0 weeks each. A system of equations is estimated every period. This provides 20 estimates from the semipooled sample, one for each period. Using the classical FamaMacBeth approach, the average slopes are the timeseries averages of the slopes obtained from the 20 regressions. The tstatistic is simply the average slope divided by its timeseries standard error across the 20 regressions. The basic system of equations is: LPREM,, Y2, Y31, Y4 = = (40) + 0i = (40 + 02i, = f41 <4,) + 0.6, where, i= to 90 bond funds and t= to 20 periods. Y3 and Y4 are some proxies that measure unobservable liquidity (4,) with error (0). These equations are estimated simultaneously as a system with P, 4P and 0 as parameters and the estimates are consistent [See, Bollen (1989) for details. The liquidity proxies are lagged by one week in the estimation process. TABLE XIII LATENT VARIABLE MOD (CONTINUED) 'ELS. AVERAGE SLOPES OF CROSSSECTIONAL ANALYSES. CLOSEDEND BOND FUNDS LISTED ON NYSE: JANUARY 1988 TO DECEMBER 199 Each row n the table ahov shows estimates of a separate model of simultaneous equations, and each column within a given row is a separate equation within the model. cell entries represent estimates of coeffi clients (or estimates ot' p) and tvalues of estimates are shown in parentheses. LPREM LVOL LDOL LSZ LSH LTURN LSIG 0.024 (7.15) 1.12 (9.55) 1.26 (7.80) 0.022 (6.39) 1.23 (13.16) 1.33 (11.61) 0.009 (1.81) 1.53 (5.59) 0.69 (7.53) 0.012 (2.54) 1.31 (5.71) 0.29 (6.74) 0.020 (6.09) 1.30 (13.16) 1.10 (13.84) 0.026 (10.00) 1.27 (5.25) 0.24 (1.35) 0.025 (7.56) 1.39 (6.05) 0.19 (8.21) 0.037 (9.00) 0.90 (5.20) 0.27 (8.07) 0.029 (9.37) 0.84 (6.81) 0.15(12. 15) 0.025 (4.61) 1.04 (4.32) 0.64 (2.89) 0.020 (4.95) 0.96 (5.12) 0.31 (3.85) 0.019 (3.98) 1.51 (18.15) 0.82 (13.92) 0.17(10.71) 0.013 (3.14) 1.55 (22.58) 1.00 (24.90) 0.16(10.16) 0.012 (3.01) 1.73 (15.07) 0.48 (9.73) 0.14(7.27) 0.022 (6.92) 1.17 (15.73) 1.01 (13.93) 0.12(5.72) 0.017 (5.37) 1.24 (42.83) 1.07 (63.83) 0.16(13.76) 0.019 (5.27) 0.86 (2.35) 0.21 (1.73) 0.11(4.03) 0.035 (9.83) 0.86 (8.85) 0.27 (7.96) 0.15(9.44) 0.020 (4.62) 1.14 (10.55) 0.40 (4.49) 0.15 (8.57) 76 TABLE XIV BINOMIAL LOGIT ANALYSIS OF THE PROBABILITY OF OBSERVING PREMIA/DISCOUNTS IN CLOSEDEND FUNDS LISTED ON THE NYSE: JANUARY 1988 TO DECEMBER Premia Discount Difference Equity Fund 0.18' 0.82* 0.64' Bond Fund 0.53' 0.47* 0.06 Difference 0.35* 0.35* * denotes significance at Probabilities of premia/discount, given fund Binomial logit specification level. ype, are shown above is used to estimate the probability of observing prem ia or discounts given the type of fund (equity or bond). The salmp Il consists of weekly observations, 'f1 1/4/88 to 12/31/91, on 18 equity lifunds and 90 bond funds are listed on the NYSE. Ilnieseries and cressscction, data is used in this analysis The basic log odds nKmodel Log (Y,/ Y,) Where, i is an index for observe tionls. = 1 for premil f 0 lor discounts. X, denotes the type of fund, I for equity fund or =2 for bond fund. = 0 +0 TABLE SENSITIVITY CLOSEDEND EQUITY FUND JANUARY ANALYSIS. LISTED ON NYSE: TO DECEMBER * SUR2 analysis not reliable hc aluse it had to bc limited it teLn weeks of dtta for the purpose identification. Small sample properties of SUR estimates are not appealing. From January 4, 1988. log of size (LSZ,,), IOg of volume (LVOL ). log of dollar volume of trade (LDOL,), shares outstanding (LSH,,), log turnover rate (LTURN,,). hlI of volatilit y (LSIG,,) and log premium (LPREM,,) are calculated for each fund on a weekly basis for 208 weeks. Five different economy it etric techniques are used to study the rohustlness of the relItionship between liquidity and premia. Fixed Effects Model (FIM), Random Effects Model (REIM), Coefficients Model (RCM) seemingly unrelated regresstions models (SURI ,SUR2) were used. dependent variable is premium (LPREM). The general llmodel i LPREM, = t,, + Y, (liquidity proxy),. where i is an index lor rund and I is an index for" Different restrictions ion paIfIrameters errors are considered by the five Models. Sec section IV for details IV. Cocfficients (B) and abIove is a stand iated tvalues Iare shown Each cell alone result. Each LVOL LDOL LSZ LSH LTURN LSIG FEM 0.010(8.33) 0.023(19.18) 0.023(16.05) 0.007(4.77) 0.023(9.30) 0.036(11.56) REM 0.010(8.39) 0.023(19.35) 0.023(16.31) 0.007(4.70) 0.022(9.44) 0.027(9.64) RCM 0.010(7.14) 0.022(16.58) 0.021(14.98) 0.006(3.85) 0.028(8.58) 0.031(6.79) SUR1 0.009(21.53) 0.022(43.98) 0.024(37.06) 0.006(9.18) 0.022(23.37) 0.004(2.81) SUR2' 0.003(2.24) 0.014(10.72) 0.008(2.68) 0.002(1.14) 0.011(9.92) 0.010(7.70) row tassoCe TABLE TIMESERIES ANALYSIS OF RELATION BETWEEN MONTHLY EXCESS RETURNS ON VOLUME DECILE PORTFOLIOS AND CLOSEDEND FUNDS LISTED ON NYSE. JANUARY 1963 TO DECEMBER Results of twelve sepanirte excess lineseries return on all closedend funds in month t ressionls is valueweighted , are shownll helow. Y, excess returns, in month I, : valueweighted on the i volume decile portfolio of the NYSE listed stocks (excluding closedend funds). The excess returns arc calculated as value weighted returns on a portfolio minus the returns on valueweighted market. The basic model = a, + 0, X,, This equation is estimated separately for each docile, i, over Ui tlimeseries of 348 months (t). Coefficients,t,, and associated tvalues (in parentheses) are shown above. Volume Decile, i /3, R % 1 (smallest volume) 0.52 (13.34) 34% 2 0.38 (9.20) 20% 3 0.34 (7.48) 14% 4 0.31 (6.37) 11% 5 0.30 (5.55) 8% 6 0.22 (3.48) 3% 7 0.31 (4.12) 5% 8 0.24 (3. 18) 3% 9 0. 11 (1.33) 0.5% 10 (largest volume) 0.08 (1.07) 0.33% Size Decile 1 0.16 (6.86) 12% Size Decile 10 0.87 (4.85) 6.4% APPENDIX A: PROPERTIES OF PORTFOLIOS PROPERTIES OF TEST PORTFOLIOS IN TERMS OF AVERAGE YEARLY RETURNS, YEARLYAVERAGE TRADING VOLUME, AVERAGE MARKET CAPITALIZATION AND THE AVERAGE NUMBER OF FIRMS JULY IN A GIVEN PORTFOLIO: 1963 TO JULY Four types of portfolios of NYSE listed firms are flnned and examined, the rules for construction of the portfolios are described below: 1) Twenty size based portfolios are ionned on July every car, chased on the market capitalization of June' 30 of the same year. Portfolio an equally weighted portfolio of the smallest of the NYSE listed finns a given year. Portfolio 20 represents an equally weighted pofolflio of the lir"est of the NYSE listed finns in vel year. The properties of these portfolios are prese ntccd in panel A. Average retlums are the time series average yearly returns (percentage change in the value of one dollar of investment) on these portf li)os over the 28 period. Average Size is the time series s average of size a portlfilio; where the a portfolio is an equally weighted average of the natural logarithm of market capitalizations of finns included in a portfolio. Average trading volume is a tlie series average of the trading volume for a portfolio; where the trading volume for weighted average of the natu nral logarithm of movingv average of past year s volume for finns included in portfolio. wenty volume based portfolios are ftormd and examined using a protcedurc similar to the one above except the basis foir finni ng portfolios volume instead i)1 S1" Volume is calculated using the past year's average trading volume a linl and this number is used to divide NYSE listed linns intt tvweinty groups. 3) Thirty sizevolume based portfolios are li 11ned every ,as follows: On July' 1 of e very year ten based portfolios ar lionned as described in I) above. Each of these ten portfolios is further subdivided into portfolios based on past year's average trading volume flor each of the member finns in a gi ven size gr iup. The breakpoints for volume are calculated by dividing all the NYSE listed finlls into three groups on the basis (oi trading a portfolio 4) Thirty volumesize based portfolios are lnired similar to the sizevolume portfolios except the firmnns first ranked on volume and then on Types of Portfolios used in Pane Panel A: Panel B: Panel C: Panel [ Panels A throu size based portfolios 20 volume based portfolios sizevolume based portft ): volumesize based portfolios A through D are on the following four separate pages. size. APPENDIX A: PANEL A SIZE BASED PORTFOLIOS. TWENTY PORTFOLIOS ARE FORMED EVERY YEAR OVER A 28 YEAR PERIOD. THE TIME SERIES AVERAGES OF PROPERTIES OF THESE PORTFOLIOS ARE SHOWN BELOW : TIME PERIOD JULY1963 TO JULY1990. Portfolio Average Returns % Avg.Ln(Size) Avg.Ln(volume) Finns I (small) 17.84 16.25 10.85 54.64 2 18.26 17.03 11.04 55.07 3 18.79 17.42 11.18 55.21 4 15.46 17.70 11.38 55.07 5 18.92 17.96 11.47 54.93 6 16.39 18.20 11.61 55.29 7 17.96 18.42 11.78 55.14 8 18.29 18.64 11.87 55.07 9 16.24 18.86 12.04 55.25 10 15.69 19.07 12.14 54.86 11 14.17 19.27 12.30 55.29 12 13.65 1949 12.49 55.21 13 14.68 19.71 12.61 55.11 14 14.75 19.95 12.77 55.14 15 14.98 20.20 12.96 55.04 16 12.36 20.43 13.12 55.14 17 13.20 20.70 13.27 55.11 18 11.81 21.00 13.43 55.21 19 10.92 21.40 13.69 55.07 82 APPENDIX A: PANEL B VOLUME BASED PORTFOLIOS. TWENTY PORTFOLIOS ARE FORMED EVERY YEAR OVER A 28 YEAR PERIOD. THE TIME SERIES AVERAGES OF PROPERTIES OF THESE PORTFOLIOS ARE SHOWN BELOW : TIME PERIOD JULY 1963 TO JULY 1990. Portfolio Average Returns Avg.Ln(Size) Av,.Ln(volume) Finns 1 (low volume) 17.63 17.47 9.71 54.64 2 17.37 17.68 10.41 55.07 3 17.72 17.87 10.77 55.25 4 18.22 8.06 11.05 55.04 5 17.32 18.26 11.28 55.04 6 16.22 18.36 11.49 55.18 7 17.14 18.53 11.69 55.18 8 15.42 18.68 11.88 55.04 9 14.79 18.86 12.07 55.25 10 15.65 19.05 12.25 54.89 11 15.21 19.24 12.43 55.25 12 16.05 19.41 12.61 55.21 13 16.29 19.67 12.79 55.14 14 13.78 19.80 12.96 55.10 15 14.09 19,93 13.14 55.04 16 14.13 20.15 13.34 55.14 17 13.55 20.34 13.56 55.21 18 12.69 20.54 13.81 55.21 19 12.49 20.82 14.12 55.07 20 (High Volume) 9.63 21.46 14.72 54.75 83 F_______________ w' S S.N oo 0 .J0. 00 '.0 ~1 __ __ ___  9en e 60 0 n W  0~L~C 9 . O w c E iN 0 0 .o0 : E N '^(NC 0 o  010 C] 00  N 7 o Lo l t1 o N ..< 3.oo < p E ^^^_ p ..Jmr^ ( S N N 0c LU LU ow^^> 7 e ~ Io n >N 0  fci 00 > 'S  '* ^ RIC;/  0 ^ If V ^  Li 4) ^ H 03 0^L  C] Cl 1? '0 00 ON ent 't> n in en i n 'oo N Sri e N n t *t? Cl * r o C' (N NC r f oN n en en en Ien en m w f *i ( i m fi Soo N oo 'toocl 100 (N' (N (N CA (NC Cl(^  ^ ^ * * "^ s 0 r(i ~ ^ ^ r'0 * 9 9 9 . ** ***  O  84 H _____________ C~v t4 In e o.  UJ~. a so b O  o4 oo ON HO a C 00 e OL ON H  ~ * N S Q^U) 6\ 000e ;o a a O 0 . L. ^ 0* D" 0" 0    O o Li J f'^,^. ^ Q 0  SJ C ^yo ^ en Cti N * a' 1...S I o oN ifl Nt 0 Cl Sr en 'C 000 > Sin r n ci "  QN In to n o 0 0 0 0 0 co  Col Cl Cl oo en * 9 9 9 9 ~ \ ^ " or * 9 9 9 9 ^ N^ N^ N~ N N^ N APPENDIX B TESTABLE HYPOTHESES: AN ALGEBRAIC ILLUSTRATION Hypothesis #1 Within a group of funds that hold similar assets (equity or bond), premia increase (discounts decrease) as the liquidity ffimunds' increases. Consider traded securities indexed on with market value per share defined as The perfect market value per share (value of cash flows promised by the security in a perfect market) is denoted by Pi. Let us write the relationship between S, and P as follows: = P Fi*P =PI*(1F)=P*L where F, is a fraction of P, that is lost due to trading friction, and L, is a measure of liquidity of security i. Equation (1) says that the market value (S,) of a security equal to the perfect market value (P,) adjusted by some liquidity measure (L,); this a statement that incorporates the value of liquidity. Next, consider a closedend fund that holds N; shares of security in its portfolio. can express the market value (MV,,,) and net asset value of the fund (NAV,,, ) in terms of the fund's holdings as follows: shares NAVV, ,C = N SS=I ZN, MV ud = Lu where equation (2) is an accounting identity and equation (3) is a restatement of equation (1) in the context of a fund. The liquidity of fund' shares is denoted by LI lund From equations (2) and (3) we can write: +premium MVf. NAV n,, Ll= where, ., is the weight of security i in the perfect market value of the fund's portfolio. The law of large numbers says that as the number of securities in the fund increases the weighted sum of liquidity the population (u). Let * L , will approach the mean liquidity of be the mean liquidity of assets of type j (equity or bond). Equation (4) suggests that within a given group of funds (equity or bond) the denominator, will be constant and premia increase as the numerator, Lrua, increases; this is a statement of hypothesis #1. Hypothesis #2: A average premia/discounts may be different across different types of funds (equity versus bond). ~ .. .. ... k  ~ f ^ \ kL_ **/ *&  ... .j r j .* ^ &L  J \,k J^n* .hk ... .' J*.l Il, .k, A .',J^. .. , .. .. ..^ f y N. P. 1 I bars denote averages. likely to be the same or L equity "fund traded in similar market structures. In this expression the average liquidity of the fund shares is = L hon, because they both are equity securities and are liund However, the mean liquidity of the bond assets is likely to different from the average liquidity of the equity assets (or I'equity assets $bond :ssets) because they are traded in different market structures and to this extent hypothesis #2 seems to be reasonable. REFERENCES Admati, Anat and Paul Pfleiderer, 1988, A theory of intraday patterns: price variability, Review of Financial Studies 1(1), 340. Volume and Amihud, Yakov, 986, Asset pricing and the bidask spread, Journal of Financial Economics 17, 223249. Amihud, Yakov and Haim Mendelson, , Liquidity, maturity and the yields on U.S. treasury securities, Journal of Finance 46 (4), 1426. Baldwin, Carli ss and Richard Meyer, Journal of Financial Economic Banz, Rolf W 979, Liquidity preference under uncertainty, 7, 347374. , The relationship between returns and market value of common stocks, Journal of Financial Economics 9, Barnea, Amir, 1974, Performance evaluation of New York Stock Exchange specialists, Journal of Financial and Quantitative Analysis 9(4), 5 11535. Barry , Christopher and Stephen Brown 1984 Differential Information and the Small firm Effect Journal of Financial Economics 13 283294. Benston, George and Robert Hagerman, 1974, Determinants of bid OverTheCounter market, Journal of Financial Economics, Iask spreads in the 1(4), 353364. Berk, Jonathan B., 1992, Does size really matter?, University of British Columbia Working Paper. Berk, Jonathan B. and Nobuya Takezawa, 1993, A reexamination of the size anomaly Bernstein, Peter, University of British Columbia Working Paper. 987, Liquidity, stock markets and market makers, Financial Management 16(2), 5462. Black, Fisher, 1986, Bollen, Kenneth A., , Noise, Journal of Finance 41(3) 529543. 1989, Structural equations with latent variables, New York: WileyInterscience, John Wiley and Sons. Boudoukh, Jacob, Govern men , Liquidity as a choice variable: A Lesson from the Japanese Bond Market, Review of Financial Studies 6, no. ,265292. Boudoukh, Jacob and Robert Whitelaw, The benchmark effect in the Japanese Government bond market, Journal of Fixed Income 1(2), 5259. Brennen, Michael, J., 1990, Latent assets, Journal of Finance 45(3), 709730. Campbell, John Sanford J. Grossman and Jiang serial correlation in stock returns, Wan working paper no. 1992, 4193 Trading volume and , NBER working paper series. Campbell John Y. and Yasushi Hamao, 992, Predictable stock returns in the United States and Japan: A study of longterm capital market integration, Journal of Finance 47(1), 4369. Chan, L. K., Hamao and J. Lakonishok, Japan, Journal of Finance, 46 (5), Fundamentals and stock returns in 17391764. Chen, NaiFu, 1983, Some empirical tests of the theory of arbitrage pricing, Journal of Finance 35 13931414. Chen, NaiFu, Raymond Kan and Merton Miller, 1993 Are the discounts on closed end funds a sentiment index, Journal of Finance 48(2), 795800. Conover, W.J. 1986 , Practical Nonparametric Statistics, Second Edition, New York: John W ey & Sons. Constar ntinides, George, 1986, Capital market equ Journal of Political Economy 94, 842862. Davidson, R. and J. MacKinnon, lilibrium with transaction costs, 1981, Several tests for model specification in the presence of alternative hypotheses, Econometrica 49, 3, 781 IT I n I 0 A cl I . I Ii ...... 1 fl I n a m ssmmore an o mo . 11 1  O * I f T Od Tni,,a ,'rlm" Diamond, Douglas W. and Phillip H. Dybvig, 1983, Bank runs, deposit insurance and liquidity, Journal of Political Economy 91, 40119. Diamond, Douglas W and Robert E. Verrecchia, , Disclosure, liquidity, and the cost of capital, Journal of Finance 46, Dimson, Elroy 13251360 , 1979, Risk measurement when shares are subject to infrequent trading, Journal of Financial Economics 197226. Fama, Eugene F. and James MacBeth, 1973 , Risk, return and equilibrium: Empirical tests, Journal of Political Economy 81 607636. Fama, Eugene F. and Kenneth R. French, 1992, The crosssection of expected stock returns, Journal of Finance 47 Flannery, Mark J. , Designing securities with market liquidity in mind, working paper, University of Florida. George, Thomas, Gautam Kaul and M. Nimalendran, Estimation of the bidask spread and its components, Review of Financia Stud 4(4), 623656. Glosten, Lawrence and Paul Milgrom, 1985 , Bid, ask and transaction prices in a specialist market with heterogeneously informed traders, Journal of Financial Economic 14(1), 71100. Goldman, M. B. and A. Beja, 1979, Market prices vs. equilibrium prices: return variance, serial correlation and the role of the specialist, Journal of Finance 34(3), 595607. Gorton , Gary, and George Pennacchi, securities. , Security baskets and indexlinked Working Paper. Philadelphia: University of Pennsylvania, Wharton School. Greene, William H., 1990 , Econometric Analysis, New York: Macmillan Publishing 461498. Grossman, Sanford J. and Merton Miller, Journal of Finance 43(3), 61 Tk',c. anti 1lane QCtnl l I V A I 1988, Liquidity and market structure, 7637. hnfri inno1 rcl^cltr nrr,;nnr itiAr traeiicartitnc anA Company Jacklin , Charles J., 1987, Demand deposits, trading restrictions and risk sharing, Stanford (mimeo). In Ed Prescott and Neil Wallace, eds., Financial Intermediation and Intertemporal Trade. Minneapolis: University of Minnesota Press. James, Christopher M. and Robert O Ed mister, The relation between common stock returns, trading activity and market value, Journal of Finance 38(4) 10751086. Jegadeesh, Narasimhan, 1992, Does market risk really explain the size effect?, Journal of Financial and Quantitative Analysis 7(3), 337351. Keim, Donald B., 1983, Size related anomalies and the stock return seasonality:Empirical evidence, Journal of Financial Economics 12, Kyle, Albert, 1315 1332. 1985, Continuous auctions and insider trading, Econolnetrica 53(6), 335. Lee, Charles M.C., Andrei Shleifer, and Richard H. Thaler, Investor sentiment and the closedend fund puzzle, Journal of Finance 46, Charles M.C ., Belinda Mucklow, and Mark Ready, 1993 , Spreads, depths and the impact of earnings information: An intraday analysis, University of Michigan. working paper Lippman, Steven A. and John J. McCall , 1986, An operational measure of liquidity, American Economic Review Thomas and Jowell S. Sabino, tests, Journal of Financial Ec 1992, Research design issues in groupingbased onomics 32, 355387. Maddala, ., and M. Nimalendran, 1993, An unobserved component panel data model to study the effect of earnings surprises on stock prices, trading volumes, and spreads, University of Florida, Working paper. Mclnish, Thomas H., and Robert A. Wood, bid/ask spreads for NYSE stocks, Join 1992, An analysis of intraday patterns in rnal of Finance 47(2), 753764. Merton, Robert C., 1987 simple model of capital market equilibrium with non ,/iv^ nln^ in n\ Cri ^itin^ In t rni 01 /C li i^nr^a' At) i Q\ A A "i_ 1in Pagano, Marco, 1989a, Trading volume and asset liquidity, Quarterly Journal of Economics, 255274. ,1989b, Endogenous market thinness and stock volatility Economic Studies 56, 269288. Review of Reinganum, Mark R. 1990 , Market microstructure and asset pricing, Journal of Financial Economics 127147 Schneikman, J. A. and L. Weiss, 1986, Borrowing constraints and aggregate economic activity, Econometrica 54, 2345. Shleifer, Andrei and Robert W Journal of Finance 4 Vishny, 1992, Liquidation values and debt capacity, 3431366. Stoll, Hans R. 1985, Alternative views of market making, in Yakov Amihud and Robert Schwartz (eds), Market Making and the Changing Structure of Security Industry, New York: Lexington Books, 6792. Stoll, Hans R. and Robert E. Whaley, 1983, Transaction costs and the small firm effect, Journal of Financial Economics Subrahmanyam, A., Financial Studies 4 (1), 5779. , A theory of trading in stock index futures, Review of 1751. Tauchen, G. and M. speculative Pitts, markets 1983, The price Econometrica variabilityvolume relationship on 1. 485505. Tinic, Seha, 1972, Economics of liquidity services, Quarterly Journal of Economics 7993. Varian, Hal R., 1988, Differences of opinion in financial markets, working paper no. 8803, The University of Michigan. BIOGRAPHICAL SKETCH Vinay Datar was born and brought up in Bombay, India. He received the National Merit scholarship, and graduated from highschool with 101' rank at the state level. He received undergraduate degree in Mechanical Engineering from the Indian Institute of Technology (I.I.T.). He had the first rank in the entrance examinations (among over a million applicants at national level) for graduate level studies at I.I.T. At the State University of New York at Buffalo, he studied Mechanical Engineering and Systems Analysis and left the graduate program before submitting the dissertation. He worked with Nashua Corporation on the Quality Program directed by Edward Demming. As a cofounder of a startup company, he raised about two million dollars in venture capital, and later sold the company to the public. At Xerox Corporation in Rochester, he was involved in Research and Development as a senior Engineer, and later he joined the Ph.D. program in Finance at the University of Florida. certify tha1 have real this study and lima acceptable standards of scholarly presentation II lily op)lnlon and is lu y adequate, conl form in scope and quality as a dissertation for degree of Doctor of P'1 losophy. 1< Robert C. ~adcli lfe Chaifhian Associate Professor of Finance certify tha have acceptable standards of scho quality, as a dissertation for ead this study and early presentation degree in mily opinion it conlforms and is fully adequate, of Doctor of in scope and iloso M iwL J. llannery Prolfessor of financee certify accepltab that I standards ave read study of scholarly presentation I thai in imy opinion it cooforms to and is fully adequate, in scope aUI quality as :I disserta on for the ree of f Doctor of 'hilosoplhy Aoilisoan H. reo [illon Associate Professor of Economnics certify that I have read this stu dy and acceptable standards of scholarly presentation that in my opinion it conforms to mnd is fully adequate, in scope and Iiality as a dissernl ion for ree of Doctor of Philosophy. c\)2 tNJr^^rvvyyv Nimalelndran island Protfessor of Finance disserta oni was he Graduate Faculty of the )Department of Finance, Insurance and Real Estate in College ol Ihiusiness Admnii the Graduate Schoo and was accepted as partial fu! C ment of he requirements for the degree of Doctor of Philosophy. sIralion and to 